Recent zbMATH articles in MSC 20https://www.zbmath.org/atom/cc/202021-02-12T15:23:00+00:00WerkzeugPermutation groups and periodicity of systems of difference equations.https://www.zbmath.org/1452.390012021-02-12T15:23:00+00:00"Hamza, Alaa E."https://www.zbmath.org/authors/?q=ai:hamza.alaa-e"Zeyada, Nasr"https://www.zbmath.org/authors/?q=ai:zeyada.nasr-a"Ahmed, A. M."https://www.zbmath.org/authors/?q=ai:ahmed.ali-mohamed|ahmed.ahmedin-mohammed|ahmed.adil-mSummary: Let \(k\in\mathbb{N}\), \(\mathbb{Z}_k=\{1,2,\dots,k\}\) and \(S_k\) be the group of all permutations on \(\mathbb{Z}_k\). Let \(\pi\in S_k\) be of order \(l\) and \(f_i\) be a function from a nonempty set \(X\) into itself, \(i=1,\dots,k\). In this paper, we show that a sufficient condition for a system of difference equations
\[
x_{n+1}^{(1)}=f_1(x_{n-s}^{(\pi(1))}),\quad x_{n+1}^{(2)}=f_2(x_{n-s}^{(\pi(2))}),\dots x^{(k)}_{n+1}=f_k(x_{n-s}^{(\pi(k))}),\,n\in\mathbb{Z}^{\leq 0},
\]
to be periodic with a period \(d\) is that each difference equation
\[
y_{n+l(s+1)-s}=g_i(y_{n-s}),\quad n\in\mathbb{Z}^{\leq 0},
\]
is periodic, \(i=1,\dots,k\), with a period \(d_i\) that divides \(d\). Here, \(g_i\) is defined by
\[
g_i=f_if_{\pi(1)}\cdots f_{\pi^{l-1}(i)},i=1,\dots,k.
\]
Finally, the periodicity of many systems of rational difference equations is established.Cross-connection structure of concordant semigroups.https://www.zbmath.org/1452.200552021-02-12T15:23:00+00:00"Azeef Muhammed, P. A."https://www.zbmath.org/authors/?q=ai:azeef-muhammed.p-a"Romeo, P. G."https://www.zbmath.org/authors/?q=ai:romeo.parackal-govindan"Nambooripad, K. S. S."https://www.zbmath.org/authors/?q=ai:nambooripad.k-s-subramonianAuthors' abstract: Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalized Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalization of the cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.
Reviewer: Peeter Normak (Tallinn)Isotropic quadrangular algebras.https://www.zbmath.org/1452.170332021-02-12T15:23:00+00:00"Mühlherr, Bernhard"https://www.zbmath.org/authors/?q=ai:muhlherr.bernhard-matthias"Weiss, Richard M."https://www.zbmath.org/authors/?q=ai:weiss.richard-mQuadrangular algebras arise in the theory of Tits quadrangles. They are anisotropic if and only if the corresponding Tits quadrangle is, in fact, a Moufang quadrangle. Anisotropic quadrangular algebras were classified in the course of classifying Moufang polygons by \textit{J. Tits} and the second author in [Moufang polygons. Berlin: Springer (2002; Zbl 1010.20017)]. A formal definition and a purely algebraic classification of quadrangular algebras were given subsequently by the second author in [Quadrangular algebras. Princeton, NJ: Princeton University Press (2006; Zbl 1129.17001)]. In this paper, the authors extend the classification of anisotropic quadrangular algebras to a classification of isotropic quadrangular algebras satisfying a natural non-degeneracy condition.
Reviewer: Anatoli Kondrat'ev (Ekaterinburg)The E-normal structure of Petrov's odd unitary groups over commutative rings.https://www.zbmath.org/1452.200472021-02-12T15:23:00+00:00"Preusser, Raimund"https://www.zbmath.org/authors/?q=ai:preusser.raimundLet \(\mathcal{V}\) be an odd quadratic space of Witt index at least \(3\) over a commutative ring with pseudoinvolution. \textit{V. A. Petrov} [J. Math. Sci., New York 130, No. 3, 4752--4766 (2005; Zbl 1144.20316); translation from Zap. Nauchn. Semin. POMI 305, 195--225 (2003)] has defined odd unitary groups \(\mathrm{U}(\mathcal{V})\). Special cases of these groups include all the classical Chevalley groups as well as the hyperbolic unitary groups of \textit{A. Bak} [The stable structure of quadratic modules. New York, NY: Columbia University (PhD Thesis) (1969)]. Let \(\mathcal{V}_0\) be the orthogonal complement of the odd quadratic space spanned by three mutually orthogonal pairs in \(\mathcal{V}\). It is known that \(\mathrm{U}(\mathcal{V})\) is isomorphic to the odd hyperbolic unitary group \(\mathrm{U}_6(\mathcal{V}_0)\).
Let \(\mathrm{EU}_6(\mathcal{V}_0)\) be the subgroup of \(\mathrm{U}_6(\mathcal{V}_0)\) generated by all its \textit{elementary transvections}. Let \(\mathfrak{F}_0\) be an \textit{odd form ideal on} \(\mathcal{V}_0\).
The normal closure of the subgroup of \(\mathrm{EU}_6(\mathcal{V}_0)\) generated by its \(\mathfrak{F}_0\)-\textit{elementary transvections} is denoted by \(\mathrm{EU}_6(\mathcal{V}_0,\mathfrak{F}_0)\). The \textit{full congruence subgroup of level \(\mathfrak{F}_0\), \(\mathrm{CU}_6(\mathcal{V}_0,\mathfrak{F}_0)\)}, is defined in a similar way to the usual definition of congruence subgroups in classical linear groups. The purpose of this paper is to prove the following sandwich classification theorem (SCT).
A subgroup \(H\) of \(\mathrm{U}_6(\mathcal{V}_0)\) is normalized by \(\mathrm{EU}_6(\mathcal{V}_0)\) if and only if
\[
\mathrm{EU}_6(\mathcal{V}_0,\mathfrak{F}_0) \subseteq H \subseteq \mathrm{CU}_6(\mathcal{V}_0,\mathfrak{F}_0)
\]
for some uniquely determined odd form ideal \(\mathfrak{F}_0\).
This adds to the list of many existing results. As with many SCT's the proof involves a great deal of intricate manipulation. A typical SCT seeks to prove, for a classical type linear group \(G\) of dimension \(n\) over a ring \(R\), that every subgroup of \(G\) with a suitably ``large'' normalizer (typically that generated by all its ``elementary'' matrices or transvections) is ``sandwiched'' between two subgroups each of which is defined by some algebraically determined subset of \(R\). The first SCT was proved by \textit{H. Bass} [Publ. Math., Inst. Hautes Étud. Sci. 22, 489--544 (1964; Zbl 0248.18025)] for the case \(G=\mathrm{GL}_n(R)\), (with some restrictions on \(R\)). Here, \(R\)-ideals are sufficient to for this SCT to hold. The next important step is due to Bak [loc. cit.] who proved an SCT for hyperbolic unitary groups. In this case, something more general is required, the so-called \textit{form ideals}.
For an SCT to hold however \(n\) must be ``sufficiently large''. For example, an SCT does hold for \(\mathrm{GL}_n(R)\), where \(R\) is any commutative ring (requiring only the \(R\)-ideals), provided \(n\geq 3\), On the other hand no SCT holds for \(\mathrm{GL}_2(\mathbb{Z})\).
Reviewer: Alexander W. Mason (Glasgow)Generators and relations of ``lower-triangular'' automorphism groups of free groups.https://www.zbmath.org/1452.200312021-02-12T15:23:00+00:00"Satoh, Takao"https://www.zbmath.org/authors/?q=ai:satoh.takaoLet $F_n$ be a free group of rank $n\geq 2$ with basis $x_1,x_2,\ldots,x_n$, $A_n=\Aut(F_n)$ the automorphism group of $F_n$,
and $\Lambda_n$ the group of integral lower-triangular matrices in $\mathrm{GL}(n,\mathbb{Z})$ whose diagonal entries are one.
For every $1\leq i\not= j\leq n$, automorphisms of $F_n$ defined by $x_i\mapsto x_i x_j$ and $x_i\mapsto x_j^{-1} x_i$
are denoted by $E_{ij}$ and $E_{i^{-1}j}$, respectively (see [\textit{W. Magnus}, Acta Math. 64, 353-367 (1935; Zbl 0012.05405)]).
Let $SA_n$ be the preimage of $\mathrm{SL}(n,\mathbb{Z})$ by the natural epimorphism $\rho:A_n\longrightarrow \Aut(F_n/[F_n,F_n])$, which is called the special automorphism group of $F_n$. Consider the subgroup $A_n^{+}$ of $SA_n$ generated by the automorphisms
$E_{ij}$ and $E_{i^{-1}j}$, for all $1\leq j<i\leq n$. The group $A_n^{+}$ is said to be the lower-triangular automorphism group of $F_n$, and it can be regarded as a free group analogue of $\Lambda_n$.
In this paper, it is given a finite presentation for $A_n^{+}$ and a normal form for its elements, using a similar technique
as that Magnus used to give the presentation for $\Lambda_n$ in the above paper.
Moreover, the kernel of $\rho$ is called the $IA$-automorphism group of $F_n$, denoted by $IA_n$, which is finitely generated by automorphisms
\[
K_{ij}:x_t\mapsto\begin{cases} x_j^{-1}x_i x_j,&t=i\\ x_t,&t\not=i \end{cases}
\]
for distinct $i,j\in \{1,2,\ldots,n\}$ and
\[
K_{ijl}:x_t\mapsto\begin{cases} x_i[x_j,x_l], &t=i\\ x_t, &t\not=i \end{cases}
\]
for distinct $i,j,l\in \{1,2,\ldots,n\}$ such that $j<l$ (see the Magnus's paper).
The author introduced the lower-triangular $IA$-automorphism group $IA_n^{+}$ in his previous paper [J. Algebra Appl. 16, No. 5, Article ID 1750099, 31 p. (2017; Zbl 1378.20050)], which is defined to be the subgroup of $IA_n$ generated by $K_{ij}$ for all $1\leq j<i\leq n$ and $K_{pqr}$ for all $1\leq r<q<p\leq n$.
In the paper under review, it is shown that $\ker (\rho|_{A_n^{+}})=IA_n^{+}$ and also it is provided an infinite presentation for $IA_n^{+}$ and a normal form for its elements.
Finally in the last section, the author computes the second homology groups $H_2(\Lambda_n,L)$ where $L$ is a principal ideal domain
in which the element 2 is invertible and $\Lambda_n$ acts trivially on $L$. This gives a lower bound on the integral second homology group of $A_n^{+}$.
Reviewer: Hesam Safa (Bojnord)Classification of regular maps of prime characteristic revisited: avoiding the Gorenstein-Walter theorem.https://www.zbmath.org/1452.050802021-02-12T15:23:00+00:00"Conder, Marston"https://www.zbmath.org/authors/?q=ai:conder.marston-d-e"Širáň, Jozef"https://www.zbmath.org/authors/?q=ai:siran.jozefSummary: \textit{A. Breda d'Azevedo} et al. [Trans. Am. Math. Soc. 357, No. 10, 4175--4190 (2005; Zbl 1065.05033)] classified the regular maps on surfaces of Euler characteristic \(- p\) for every prime \(p\). This classification relies on three key theorems, each proved using the highly non-trivial characterisation of finite groups with dihedral Sylow 2-subgroups, due to \textit{D. Gorenstein} and \textit{J. H. Walter} [J. Algebra 2, 85--151, 218--270 (1965; Zbl 0192.11902); ibid. 2, 354--393 (1965; Zbl 0192.12001)]. Here we give new proofs of those three facts (and hence the entire classification) using somewhat more elementary group theory, without referring to the Gorenstein-Walter theorem.Lower central and derived series of semi-direct products, and applications to surface braid groups.https://www.zbmath.org/1452.200332021-02-12T15:23:00+00:00"Guaschi, John"https://www.zbmath.org/authors/?q=ai:guaschi.john"de Miranda e Pereiro, Carolina"https://www.zbmath.org/authors/?q=ai:de-miranda-e-pereiro.carolinaIn the first part of the present paper, the authors give a general description of lower central series and derived series an arbitrary semi-direct product. In the second part of the paper, they study these series for the full braid group \(B_n(M)\) and pure braid group \(P_n(M)\) of a compact surface \(M\), orientable or non-orientable, the aim being to determine the values of \(n\) for which \(B_n(M)\) and \(P_n(M)\) are residually nilpotent or residually soluble.
They first solve this problem in the case where \(M\) is the 2-torus \(\mathbb{T}\). The authors prove (Theorem 1.2) that the braid group \(B_n(\mathbb{T})\) is residually soluble if and only if \(n \leq 4\).
For the braid group of the Klein bottle is proved that \(P_n(\mathbb{K})\) is residually nilpotent for all \(n \geq 1\) (see Theorem 1.3).
In Theorem 1.4 is proved that if \(M\) is a compact non-orientable surface of genus \(g \geq 1\) without boundary, then \(B_n(M)\) is residually nilpotent if and only if \(n \leq 2\), and is residually soluble if and only if \(n \leq 4\).
Reviewer: Valeriy Bardakov (Novosibirsk)Central stability for the homology of congruence subgroups and the second homology of Torelli groups.https://www.zbmath.org/1452.180032021-02-12T15:23:00+00:00"Miller, Jeremy"https://www.zbmath.org/authors/?q=ai:miller.jeremy-a"Patzt, Peter"https://www.zbmath.org/authors/?q=ai:patzt.peter"Wilson, Jennifer C. H."https://www.zbmath.org/authors/?q=ai:wilson.jennifer-c-hCet article fournit des renseignements qualitatifs sur la structure stable de l'homologie de sous-groupes de congruence de groupes linéaires de taille \(n\) sur un anneau \(A\) (congruence associée à un idéal bilatère \(I\)), du deuxième groupe d'homologie du sous-groupe \(IA_n\) du groupe des automorphismes d'un groupe libre de rang \(n\) formé des automorphismes induisant l'identité sur l'abélianisation, ou du deuxième groupe d'homologie du groupe de Torelli \(\mathcal{I}_n\) d'une surface de genre \(n\) avec une composante de bord. Ici, \textit{stable} signifie qu'on s'intéresse au comportement de ces homologies lorsque \(n\) grandit. Plus précisément, les auteurs montrent que ces suites de groupes d'homologie ont leurs valeurs déterminées, pour \(n\) assez grand (avec une borne explicite), par leurs valeurs aux entiers strictement inférieurs, où l'on tient compte de la structure fonctorielle globale, c'est-à-dire des morphismes de transition (permettant d'accroître la valeur de \(n\)) et de l'action d'un sous-groupe remarquable du groupe linéaire \(GL_n(A/I)\) dans le premier cas, du groupe linéaire \(GL_n(\mathbb{Z})\) dans le second, et du groupe symplectique \(Sp_{2n}(\mathbb{Z})\) dans le dernier. Dans le cas des groupes de congruence, on a besoin que l'anneau \(A\) possède un rang stable de Bass fini.
Les méthodes reposent sur l'utilisation de catégories de foncteurs appropriées et s'inscrivent dans la lignée de nombreux travaux, notamment les articles relatifs à la stabilité homologique [\textit{R. Charney}, Commun. Algebra 12, 2081--2123 (1984; Zbl 0542.20023)], [\textit{A. Putman}, Invent. Math. 202, No. 3, 987--1027 (2015; Zbl 1334.20045)] et [\textit{O. Randal-Williams} and \textit{N. Wahl}, Adv. Math. 318, 534--626 (2017; Zbl 1393.18006)], ainsi que les outils d'homologie des foncteurs étudiés dans [\textit{T. Church} and \textit{J. Ellenberg}, Geom. Topol. 21, No. 4, 2373--2418 (2017; Zbl 1371.18012)], [\textit{A. Putman} and \textit{S. Sam}, Duke Math. J. 166, No. 13, 2521--2598 (2017; Zbl 1408.18003)] ou [\textit{P. Patzt}, Math. Z. 295, No. 3--4, 877--916 (2020; Zbl 1442.18004)]. Des techniques simpliciales classiques sont largement employées, ainsi que la notion de \textit{foncteur polynomial}.
Malheureusement, les méthodes utilisées ne semblent pas du tout suffisantes pour donner des renseignements sur les groupes d'homologie \(H_i(IA_n)\) (ou \(H_i(\mathcal{I}_n)\)) pour \(i>2\). Il convient toutefois de signaler que les résultats donnés pour \(H_2(IA_n)\) sont hautement non triviaux étant donné la difficulté considérable à comprendre les groupes \(IA_n\) et leur homologie (on sait par exemple que \(H_2(IA_3;\mathbb{Q})\) est de dimension infinie, mais on ignore même si le groupe \(IA_n\) est de présentation finie pour \(n>3\)).
Reviewer: Aurelien Djament (Villeneuve d'Ascq)Transitivity of the \(\delta^n\)-relation in hypergroups.https://www.zbmath.org/1452.200642021-02-12T15:23:00+00:00"Ghiasvand, Peyman"https://www.zbmath.org/authors/?q=ai:ghiasvand.peyman"Mirvakili, Saeed"https://www.zbmath.org/authors/?q=ai:mirvakili.saeedSummary: The \(\delta^n\)-relation was introduced by [\textit{V. Leoreanu-Fotea} et al., Commun. Algebra 40, No. 10, 3597--3608 (2012; Zbl 1264.20077)]. In this article, we introduce the concept of \(\delta^n\)-heart of a hypergroup and we determine necessary and sufficient conditions for the relation \(\delta^n\) to be transitive. Moreover, we determine a family \(P_\sigma(H)\) of subsets of a hypergroup \(H\) and we give sufficient conditions such that the geometric space \((H,P_\sigma(H))\) is strongly transitive and the relation \(\delta^n\) is transitive.Isotopes of octonion algebras, \(\mathbf{G}_{2}\)-torsors and triality.https://www.zbmath.org/1452.170312021-02-12T15:23:00+00:00"Alsaody, S."https://www.zbmath.org/authors/?q=ai:alsaody.seidon"Gille, P."https://www.zbmath.org/authors/?q=ai:gille.philippe|gille.phillipeGiven a field \(k\), or more generally a local ring, two octonion algebras over \(k\) are isomorphic if and only if their quadratic forms are isometric. This was proved to be false over more general commutative rings by \textit{P. Gille} [Can. Math. Bull. 57, No. 2, 303--309 (2014; Zbl 1358.11126)].
Let \(C\) be an octonion algebra over a unital commutative ring \(R\), with norm \(q\). The paper under review deals with those octonion algebras over \(R\) whose norm is isometric to \(q\). The main result shows that, up to isomorphism, these octonion algebras are the isotopes \(C^{a,b}\) for norm \(1\) elements \(a,b\in C\), which are defined on \(C\), with the same norm, but with new multiplication \(x*y=(xa)(by)\).
The key ingredient to achieving this is triality. The group scheme \(\mathbf{RT}(C)\), whose \(S\)-points are the triples
\((t_1,t_2,t_3)\in \mathbf{SO}(q)(S)\) such that \(t_1(xy)=\overline{t_2(\bar x)}\,\overline{t_3(\bar y)}\) for any \(S\)-ring \(T\) and any \(x,y\in C_T\), has cyclic symmetry, and it is isomorphic to \(\mathbf{Spin}(q)\). The group scheme \(\mathbf{RT}(C)\) acts naturally on two copies of the unit sphere: \(\mathbf{S}_C^2\), and the stabilizer of \((1,1)\) is isomorphic to the automorphism group scheme \(\mathbf{Aut}(C)\). Moreover, the fppf quotient sheaf \(\mathbf{RT}(C)/\mathbf{Aut}(C)\) turns out to be isomorphic to \(\mathbf{S}_C^2\). This gives a very precise description of the \(\mathbf{Aut}(C)\)-torsor \(\Pi:\mathbf{Spin}(q)\rightarrow\mathbf{Spin}(q)/\mathbf{Aut}(C)\).
It is shown that the \(\Pi\)-twists of \(C\) correspond canonically to the isotopes \(C^{a,b}\) above, and the main result follows by proving that the torsor \(\Pi\) gives the same objects as \(\mathbf{O}(q)\rightarrow \mathbf{O}(q)/\mathbf{Aut}(C)\).
From this general framework, new results are deduced for some particular rings, like the rings of (Laurent) polynomials.
Reviewer: Alberto Elduque (Zaragoza)Gelfand-Kirillov dimension of the quantized algebra of regular functions on homogeneous spaces.https://www.zbmath.org/1452.160232021-02-12T15:23:00+00:00"Chakraborty, Partha Sarathi"https://www.zbmath.org/authors/?q=ai:chakraborty.partha-sarathi"Saurabh, Bipul"https://www.zbmath.org/authors/?q=ai:saurabh.bipulSummary: In this article, we prove that the Gelfand-Kirillov dimension of the quantized algebra of regular functions on certain homogeneous spaces of types \(A\), \(C\), and \(D\) is equal to the dimension of the homogeneous space as a real differentiable manifold.A look at representations of \(\mathrm{SL}_{2}(\mathbb{F}_{q})\) through the lens of size.https://www.zbmath.org/1452.200432021-02-12T15:23:00+00:00"Gurevich, Shamgar"https://www.zbmath.org/authors/?q=ai:gurevich.shamgar"Howe, Roger"https://www.zbmath.org/authors/?q=ai:howe.roger-eSummary: How to study a nice function on the real line? The physically motivated Fourier theory technique of harmonic analysis is to expand the function in the basis of exponentials and study the meaningful terms in the expansion. Now, suppose the function lives on a finite non-commutative group \(G\), and is invariant under conjugation. There is a well-known analog of Fourier analysis, using the irreducible characters of \(G\). This can be applied to many functions that express interesting properties of \(G\). To study these functions one wants to know how the different characters contribute to the sum? In this note we describe the \(G=\mathrm{SL}_{2}(\mathbb{F}_{q})\) case of the theory we have been developing in recent years which attempts to give a fairly general answer to the above question for finite classical groups. The irreducible representations of \(\mathrm{SL}_{2}(\mathbb{F}_{q})\) are ``well known'' for a very long time [\textit{G. Frobenius}, Berl. Ber. 1896, 985--1021 (1896; JFM 27.0092.01); \textit{H. E. Jordan}, Am. J. Math. 29, 387--405 (1907; JFM 38.0175.01); \textit{I. Schur}, J. Reine Angew. Math. 132, 85--137 (1907; JFM 38.0174.02)] and are a prototype example in many introductory courses on the subject. We are happy that we can say something new about them. In particular, it turns out that the representations that were considered as ``anomalous'' in the ``old'' point of view (known as the ``philosophy of cusp forms'') are the building blocks of the current approach.On the intersection of nilpotent Hall \(\pi \)-subgroups.https://www.zbmath.org/1452.200112021-02-12T15:23:00+00:00"Jin, Ping"https://www.zbmath.org/authors/?q=ai:jin.ping"Yang, Yong"https://www.zbmath.org/authors/?q=ai:yang.yongLet \(G\) be a \(\pi\)-separable group with a nilpotent Hall \(\pi\)-subgroup and assume that the following conditions hold: if there are Fermat primes or Mersenne primes dividing \(|G|\) but not \(|H|\), then \(H\) does not involves \(Z_2\wr Z_2\); if \(|G|\) is even and \(|H|\) is odd, then there is no Mersenne prime \(p\) such that \(H\) involves \(Z_p\wr Z_p\). Under these assumptions, for any collection of Hall \(\pi\)-subgroups \(H_1,\dots,H_n\) of \(G\), there exists an element \(g \in G\) such that \(H \cap H_1\cap \dots \cap H_n = H \cap H^g\). This generalizes a theorem of \textit{G. R. Robinson} [Proc. Am. Math. Soc. 90, 21--24 (1984; Zbl 0505.20015)] and a result of \textit{B. Brewster} and \textit{P. Hauck} [J. Algebra 206, No. 1, 261--292 (1998; Zbl 0932.20018)] giving a unified proof.
Reviewer: Andrea Lucchini (Padova)Topological folding on the chaotic projective spaces and their fundamental group.https://www.zbmath.org/1452.510032021-02-12T15:23:00+00:00"Abu-Saleem, M."https://www.zbmath.org/authors/?q=ai:abu-saleem.mohammed"Al-Omeri, W. Faris."https://www.zbmath.org/authors/?q=ai:al-omeri.wadei-farisSummary: In this article we will introduce different types of topological foldings on chaotic projective space. The limit of topological foldings on the fundamental group of the real projective plane will be obtained. The chain of folding on the chaotic projective spaces will induce a chain of fundamental groups. The relations between these chains will be achieved.Some remarks on finitarily approximable groups.https://www.zbmath.org/1452.200252021-02-12T15:23:00+00:00"Nikolov, Nikolay"https://www.zbmath.org/authors/?q=ai:nikolov.nikolay"Schneider, Jakob"https://www.zbmath.org/authors/?q=ai:schneider.jakob"Thom, Andreas"https://www.zbmath.org/authors/?q=ai:thom.andreas-bertholdSummary: The concept of a \(\mathcal{C}\)-approximable group, for a class of finite groups \(\mathcal{C}\), is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. In a related note, we prove that any non-trivial group which can be approximated by finite groups has a non-trivial quotient that can be approximated by finite projective special linear groups. Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length function. We prove that these are precisely the abelian ones, providing a negative answer to a question of \textit{M. Doucha} [Groups Geom. Dyn. 12, No. 2, 615--636 (2018; Zbl 1407.22001)]. Referring to a problem of \textit{B. Zilber} [in: Infinity and truth. Based on talks given at the workshop, Singapore, July 25--29, 2011. Hackensack, NJ: World Scientific. 199--223 (2014; Zbl 1321.03052)], we show that the identity component of a Lie group, whose topology is generated by an invariant length function and which is an abstract quotient of a product of finite groups, has to be abelian. Both of these last two facts give an alternative proof of a result of \textit{A. M. Turing} [Ann. Math. (2) 39, 105--111 (1938; JFM 64.1094.01)]. Finally, we solve a conjecture of \textit{A. Pillay} [Fundam. Math. 236, No. 2, 193--200 (2017; Zbl 1420.03063)] by proving that the identity component of a compactification of a pseudofinite group must be abelian as well. All results of this article are applications of theorems on generators and commutators in finite groups by the the first author and \textit{D. Segal} [Invent. Math. 190, No. 3, 513--602 (2012; Zbl 1268.20031)]. In Section 4 we also use results of \textit{M. W. Liebeck} and \textit{A. Shalev} on bounded generation in finite simple groups [Ann. Math. (2) 154, No. 2, 383--406 (2001; Zbl 1003.20014)].Groups with all subgroups permutable or soluble of finite rank.https://www.zbmath.org/1452.200262021-02-12T15:23:00+00:00"Dixon, M. R."https://www.zbmath.org/authors/?q=ai:dixon.martyn-russell"Ferrara, M."https://www.zbmath.org/authors/?q=ai:ferrara.maria"Karatas, Z. Y."https://www.zbmath.org/authors/?q=ai:karatas.z-yalcin"Trombetti, M."https://www.zbmath.org/authors/?q=ai:trombetti.marcoA subgroup \(H\) is permutable in the group \(G\) if \(HK=KH\) for every subgroup \(K\) of \(G\) and \(G\) is \textit{quasihamiltonian} if every subgroup of \(G\) is permutable. A group is locally graded if every nontrivial finitely generated subgroup has a nontrivial finite image.
In this paper the authors study the class of locally graded groups all of whose subgroups are permutable or are soluble and satisfy a certain rank condition. The rank conditions in question include groups of finite abelian section rank, minimax groups and polycyclic groups. In each case necessary and sufficient conditions are given for a locally graded group to have all subgroups permutable or soluble with the given rank condition.
Reviewer: Egle Bettio (Venezia)On stability of exactness properties under the pro-completion.https://www.zbmath.org/1452.180052021-02-12T15:23:00+00:00"Jacqmin, Pierre-Alain"https://www.zbmath.org/authors/?q=ai:jacqmin.pierre-alain"Janelidze, Zurab"https://www.zbmath.org/authors/?q=ai:janelidze.zurabThis paper is concerned with the question which properties of the given category carry over to its pro-completion. If \(C\)\ is a small finitely complete category, its pro-completion is no other than its free cofiltered limit completion, which is given by the restricted Yoneda embedding
\[
C\hookrightarrow\mathbf{Lex}(C,\mathbf{Set})^{\mathrm{op}}
\]
where \(\mathbf{Lex}(C,\mathbf{Set})\) is the category of finite limit preserving functors from \(C\)\ to \(\mathbf{Set}\). Many so-called \textit{exactness properties} have been shown to be stable under this construction, say, being regular [\textit{M. Barr}, J. Pure Appl. Algebra 41, 113--137 (1986; Zbl 0606.18004)], coregular [\textit{B. Day} and \textit{R. Street}, J. Pure Appl. Algebra 58, No. 3, 227--233 (1989; Zbl 0677.18009)], additive [\textit{B. Day} and \textit{R. Street}, J. Pure Appl. Algebra 63, No. 3, 225--229 (1990; Zbl 0705.18006)], abelian [loc. cit.], exact Mal'tsev with pushouts [\textit{F. Borceux} and \textit{M. C. Pedicchio}, J. Pure Appl. Algebra 135, No. 1, 9--22 (1999; Zbl 0926.18003)], coregular co-Mal'tsev [\textit{M. Gran} and \textit{M. C. Pedicchio}, Theory Appl. Categ. 8, 1--15 (2001; Zbl 0970.18008)], coextensive with pushouts [\textit{A. Carboni} et al., J. Pure Appl. Algebra 161, No. 1--2, 65--90 (2001; Zbl 0982.18006)] and extensive [loc. cit.].
The principal objective in this paper consisting of six sections is to establish a general stability theorem (Theorem 2.2) subsuming not only the said ones but also properties of being semi-abelian, regular, Mal'tsev, coherent with finite products and so on. In some sense, the authors' approach to establishing the general stability theorem is analogous to the approach exploited in the particular said cases, though the generality brings in heavy technicalities, which are tackled in use of \(2\)-categorical calculus of natural transformations. Indeed, the proof of Theorem 2.2 is relegated to \S 4 after preliminaries for the proof of the stability theorem in \S 3, being divided into 29 steps. As is expected, the authors make use of a generalization of the set-based case of a lemma from [\textit{B. Day} and \textit{R. Street}, J. Pure Appl. Algebra 58, No. 3, 227--233 (1989; Zbl 0677.18009)] called the \textit{uniformity lemma}, whose proof occupies a substantial part in the proof of the general stability theorem, relying on classical results regarding pro-completion [\textit{P. Gabriel} and \textit{F. Ulmer}, Lokal präsentierbare Kategorien. (Locally presentable categories). Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0225.18004); \textit{M. Artin} et al., Séminaire de géométrie algébrique du Bois-Marie 1963--1964. Théorie des topos et cohomologie étale des schémas. (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne, B. Saint-Donat. Tome 1: Théorie des topos. Exposés I à IV. 2e éd. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0234.00007)].
The authors' approach to formalizing the definition of an exactness property is based on the theory of \textit{sketches} due to \textit{C. Ehresmann} [Bul. Inst. Politeh. Iaşi, N. Ser. 14(18), No. 1--2, 1--14 (1968; Zbl 0196.03102)].
The present approach to exactness properties is not intended to cover all properties of a category of interest, being to be thought of as formalization of the so-called \textit{first-order} exactness properties so that such a higher-order exactness property as existence of enough projectives is out of consideration. It should be noted that the authors' stability theorem claims that not all but only certain first-order exactness properties are stable under the pro-completion, a notably exception being the exactness property of a morphism to be the truth morphism for a subobject classifier. The stability theorem claims under certain conditions that, given a functorial verification of \(\alpha\vdash\beta\) for an \(\mathcal{X}\)-structure \(F\) in \(C\), there exists a functorial verification of \(\alpha\vdash\beta\)\ for the image \(\mathbf{y}F\) of \(F\) under the (restricted) Yoneda embedding
\[
\mathbf{y}:C\hookrightarrow\mathbf{Lex}(C,\mathbf{Set})^{\mathrm{op}}
\]
Furthermore, the functorial verification of \(\alpha\vdash\beta\)\ for \(\mathbf{y}F\) can be chosen so that it is \textit{coherent} with the functorial verification of \(\alpha\vdash\beta\) for \(F\), as addressed in \S 5.
\S 6 is concerned with concluding remarks.
Reviewer: Hirokazu Nishimura (Tsukuba)Finite groups with small centralizers of word-values.https://www.zbmath.org/1452.200182021-02-12T15:23:00+00:00"Detomi, Eloisa"https://www.zbmath.org/authors/?q=ai:detomi.eloisa"Morigi, Marta"https://www.zbmath.org/authors/?q=ai:morigi.marta"Shumyatsky, Pavel"https://www.zbmath.org/authors/?q=ai:shumyatsky.pavelLet \(p\) be a prime number and \(n\) be an integer \(\geq2\) not divided by \(p\). If \(H\) is any finite abelian group of exponent \(n\), we call \(B\) the base group of the standard wreath product \(H \wr K\), where \(K\simeq\mathbb{Z}_p\). It is easy to see that in the semidirect pro\-duct \(G\) of \([B,\, K]\) by \(K\) all subgroups of order \(p\) are self-centralizing.
If we let \(w\) be the group-word \(x^n\), then the groups constructed above are examples of finite groups of \textit{arbitrary large} orders in which \(w(G)\neq1\) and all centralizers \(C_G(x)\) of non-trivial \(w\)-values \(x\) of \(G\) have order bounded by \(p\) (Remark 2). Here the assumption \(n\geq2\) for the word \(x^n\) is a crucial one for the existence of such examples; indeed, it is well-known that if all non-trivial elements of a finite group have centralizers of order at most \(m\), then the order of the group is bounded by a function of \(m\).
The aim of the paper under review is to show that there are many other words \(w\) for which a bound (in terms of \(w\) and \(m\)) is possible for a finite group \(G\) in which \(w(G)\neq1\) and all centralizers of non-trivial \(w\)-values have order at most \(m\): this for instance the case of multilinear commutator words of weight at least \(2\) (Theorem 1) and the \(n\)th Engel word for all \(n\) (Theorem 2).
Reviewer: Marco Trombetti (Napoli)Extracting a \(\Sigma \)-Mal'tsev \((\Sigma \)-protomodular) structure from a Mal'tsev (protomodular) subcategory.https://www.zbmath.org/1452.180022021-02-12T15:23:00+00:00"Bourn, Dominique"https://www.zbmath.org/authors/?q=ai:bourn.dominiqueSummary: We give conditions on an inclusion \({\mathbb{C}}\hookrightarrow{\mathbb{D}}\) where \({\mathbb{C}}\) is a Mal'tsev (resp. protomodular) subcategory in order to produce on \({\mathbb{D}}\) a partial \(\Sigma \)-Mal'tsev (resp. \( \Sigma \)-protomodular) structure.The equivalence of two dichotomy conjectures for infinite domain constraint satisfaction problems.https://www.zbmath.org/1452.080012021-02-12T15:23:00+00:00"Barto, Libor"https://www.zbmath.org/authors/?q=ai:barto.libor"Kompatscher, Michael"https://www.zbmath.org/authors/?q=ai:kompatscher.michael"Olšák, Miroslav"https://www.zbmath.org/authors/?q=ai:olsak.miroslav"Van Pham, Trung"https://www.zbmath.org/authors/?q=ai:pham.trung-van"Pinsker, Michael"https://www.zbmath.org/authors/?q=ai:pinsker.michaelOn free subsemigroups of associative algebras.https://www.zbmath.org/1452.200602021-02-12T15:23:00+00:00"Letzter, Edward S."https://www.zbmath.org/authors/?q=ai:letzter.edward-sAuthor's abstract: In 1992, following earlier conjectures of \textit{A. Lichtman} [Proc. Am. Math. Soc. 63, 15--16 (1977; Zbl 0352.20026)] and \textit{L. Makar-Limanov} [Proc. Am. Math. Soc. 91, 189--191 (1984; Zbl 0512.16018)], \textit{A. A. Klein} [Proc. Am. Math. Soc. 116, No. 2, 339--341 (1992; Zbl 0777.16017)] conjectured that a noncommutative domain must contain a free, multiplicative, noncyclic subsemigroup. He verified the conjecture when the center is uncountable. In this note, we consider the existence (or not) of free subsemigroups in associative \(k\)-algebras \(R\), where \(k\) is a field not algebraic over a finite subfield. We show that \(R\) contains a free noncyclic subsemigroup in the following cases: (1) \(R\) satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) \(R\) has at least one nonartinian primitive subquotient. (3) \(k\) is uncountable and \(R\) is noncommutative modulo its Jacobson radical. In particular, (1) and (2) verify Klein's conjecture for numerous well-known classes of domains, over countable fields, not covered in the prior literature.
Reviewer: Eric Jespers (Brüssel)Ideal theory in ordered AG-groupoids based on double framed soft sets.https://www.zbmath.org/1452.060172021-02-12T15:23:00+00:00"Asif, Tauseef"https://www.zbmath.org/authors/?q=ai:asif.tauseef"Yousafzai, Faisal"https://www.zbmath.org/authors/?q=ai:yousafzai.faisal"Khan, Asghar"https://www.zbmath.org/authors/?q=ai:khan.asghar"Hila, Kostaq"https://www.zbmath.org/authors/?q=ai:hila.kostaqSummary: In this note, we investigate the relationship between DFS-L-ideal, DFSR- ideal, DFS-2S-ideal and DFS-I-ideal of an ordered AG-groupoid over a universe by providing examples/counter examples. We characterize a right regular class of an ordered AG-groupoid in terms of these DFS-ideals.HX-groups associated with the dihedral group \(D_n\).https://www.zbmath.org/1452.200652021-02-12T15:23:00+00:00"Sonea, Andromeda Cristina"https://www.zbmath.org/authors/?q=ai:sonea.andromeda-cristinaSummary: In this paper, we determine the hypergroups associated with the HX-groups of dihedral group \(D_n\). Also we calculate the fuzzy grade of these hypergroups and the commutativity degree for the corresponding HX-groups.Very special algebraic groups.https://www.zbmath.org/1452.140452021-02-12T15:23:00+00:00"Brion, Michel"https://www.zbmath.org/authors/?q=ai:brion.michel"Peyre, Emmanuel"https://www.zbmath.org/authors/?q=ai:peyre.emmanuelSummary: We say that a smooth algebraic group \(G\) over a field \(k\) is very special if for any field extension \(K/k\), every \(G_K\)-homogeneous \(K\)-variety has a \(K\)-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions.A note on the rigidity of convergence of Möbius transformations.https://www.zbmath.org/1452.300222021-02-12T15:23:00+00:00"Lascurain, Antonio"https://www.zbmath.org/authors/?q=ai:lascurain.antonio"Nicolás, Francisco"https://www.zbmath.org/authors/?q=ai:nicolas.franciscoSummary: It is proved using elementary techniques that if a given sequence of Möbius transformations acting in \(\widehat{\mathbb{R}}^n\) converges pointwise, and for three different points in \(\widehat{\mathbb{R}}^n\) the sequence converges to three different points in \(\widehat{\mathbb{R}}^n\), one has that the sequence converges to a Möbius transformation (Theorem 3.1). Moreover, it is proved that the number three is the best lower bound.
For the entire collection see [Zbl 1429.53001].Contemporary abstract algebra. 10th edition.https://www.zbmath.org/1452.000012021-02-12T15:23:00+00:00"Gallian, Joseph A."https://www.zbmath.org/authors/?q=ai:gallian.joseph-aPublisher's description: The 10th edition of the book is primarily intended for an abstract algebra course whose main purpose is to enable students to do computations and write proofs. Gallian's text stresses the importance of obtaining a solid introduction to the traditional topics of abstract algebra, while at the same time presenting it as a contemporary and very much an active subject which is currently being used by working physicists, chemists, and computer scientists.
See the review of the 2nd edition in [Zbl 0777.00002]. For the 4th, 5th and 9th editions see [Zbl 0972.00001; Zbl 1051.00001; Zbl 1330.00006]. For the 2011 reprint of the 4th edition see [Zbl 1232.00001].Biplanes with a prime number of points and a flag-transitive automorphism group of one-dimensional affine type.https://www.zbmath.org/1452.050102021-02-12T15:23:00+00:00"García-Vázquez, Patricio Ricardo"https://www.zbmath.org/authors/?q=ai:garcia-vazquez.patricio-ricardoSummary: We study the open case of biplanes having a prime number of points \(p\) and a flag-transitive automorphism group of affine type. It is conjectured that the only example is the known flag-transitive \((37,9,2)\) biplane. We prove that this is true when \(p<10^7\).
For the entire collection see [Zbl 1429.53001].Conjugacy of Levi subgroups of reductive groups and a generalization to linear algebraic groups.https://www.zbmath.org/1452.200462021-02-12T15:23:00+00:00"Solleveld, Maarten"https://www.zbmath.org/authors/?q=ai:solleveld.maartenLet \(G\) be a connected reductive linear algebraic group over a field \(K\). The author defines a Levi \(K\)-subgroup of \(G\) to be a Levi factor of a parabolic \(K\)-subgroup of \(G\). The author compares the known rational conjugacy classes of Levi subgroups over \(K\) to their geometric conjugacy class counterparts, those over the algebraic closure \(\overline{K}\). The author does so by identifying standard Levi \(K\)-subgroups \(\mathcal{L}_{I_K}\) for each subset \(I_K \subset \Delta_K\), where \(\Delta_K\) is a simple system of roots of \(G\) with respect to a maximal \(K\)-split torus \(S\). Taken from the paper:
Theorem A: Let \(G\) be a connected reductive \(K\)-group. Every Levi \(K\)-subgroup of \(G\) is \(G(K)\)-conjugate to a standard Levi \(K\)-subgroup. For two standard Levi \(K\)-subgroups \(\mathcal{L}_{I_K}\) and \(\mathcal{L}_{J_K}\) the following are equivalent:
\begin{itemize}
\item[i)] \(I_K\) and \(J_K\) are associate under the Weyl group \(W(G,S)\);
\item[ii)] \(\mathcal{L}_{I_K}\) and \(\mathcal{L}_{J_K}\) are \(G(K)\)-conjugate;
\item[iii)] \(\mathcal{L}_{I_K}\) and \(\mathcal{L}_{J_K}\) are \(G(\overline{K})\)-conjugate.
\end{itemize}
The main results of the paper are about a generalization of Theorem A to arbitrary connected linear algebraic groups, replacing Levi subgroups with pseudo-Levi subgroups. Taken from the paper:
Theorem B: Let G be a connected linearly algebraic \(K\)-group. Every pseudo-Levi subgroup of \(G\) is \(G(K)\)-conjugate to a standard pseudo-Levi subgroup. For two standard pseudo-Levi \(K\)-subgroups \(\mathcal{L}_{I_K}\) and \(\mathcal{L}_{J_K}\) the following are equivalent:
\begin{itemize}
\item[i)] \(I_K\) and \(J_K\) are associate under the Weyl group \(W(G,S)\);
\item[ii)] \(\mathcal{L}_{I_K}\) and \(\mathcal{L}_{J_K}\) are \(G(K)\)-conjugate;
\item[iii)] \(\mathcal{L}_{I_K}\) and \(\mathcal{L}_{J_K}\) are \(G(\overline{K})\)-conjugate.
\end{itemize}
Section 2 of the paper is primarily a proof of Theorem A. The majority of Theorem A is contained in Lemma 1. The equivalence of the second and third points is the content of Theorem 2. Its proof proceeds by first showing that if the result hold for quasi-split \(K\)-groups, then it holds for all reductive \(K\)-groups. Second, it is shown that if the result holds for all absolutely simple, quasi-split \(K\)-groups, then it holds for all quasi-split reductive \(K\)-groups. Finally, working casewise by root system type, it is shown that the result holds when \(G\) is absolutely simple and quasi-split over \(K\).
In Section 3, the author works with a connected linear algebraic \(K\)-group \(G\) that is no longer assumed to be reductive. Lemma 8 is then the generalization of Lemma 1, containing the majority of Theorem B. The equivalence of the second and third points again requires considerable work and is the content of Theorem 10. The proof proceeds in a similar manner as the proof of Theorem 2. First, it is shown that if the result holds for pseudo-simple \(K\)-groups with trivial scheme-theoretic center, then it holds in full. Second, it is shown that the result holds when \(G\) is quasi-split over \(K\). Finally, using the previous quasi-split case, it is shown that the result holds for pseudo-simple \(K\)-groups with trivial scheme-theoretic center.
Reviewer: Cameron Ruether (Ottawa)The action of \(\mathrm{SL}(2,{\mathbb {C}})\) on hyperbolic 3-space and orbital graphs.https://www.zbmath.org/1452.200482021-02-12T15:23:00+00:00"Beşenk, Murat"https://www.zbmath.org/authors/?q=ai:besenk.muratSummary: In this paper we discuss the action of \(\mathrm{SL}(2,{\mathbb {C}})\) on hyperbolic 3-space using quaternions. And then we investigate suborbital graphs for the special subgroup of \(\mathrm{PSL}(2,\mathbb {C})\). We point out the relation between elliptic elements and circuits in graphs. Results obtained by the method used are important because they mean that suborbital graphs have a potential to explain signature problems.Structure of symmetry group of some composite links and some applications.https://www.zbmath.org/1452.570072021-02-12T15:23:00+00:00"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.22|liu.yang.23|liu.yang.21|liu.yang.13|liu.yang.3|liu.yang.4|liu.yang.9|liu.yang.10|liu.yang|liu.yang.2|liu.yang.20|liu.yang.18|liu.yang.12|liu.yang.19|liu.yang.6|liu.yang.14|liu.yang.8|liu.yang.16|liu.yang.11|liu.yang.15|liu.yang.5|liu.yang.7|liu.yang.1|liu.yang.17This paper considers the symmetry groups of topological links, primarily in three space. The link \(2^{2}_{1}m\#2^{2}_{1}\) is used as an example for the methods of the paper.
Reviewer: Jonathan Hodgson (Swarthmore)Fixed points of diffeomorphisms on nilmanifolds with a free nilpotent fundamental group.https://www.zbmath.org/1452.550012021-02-12T15:23:00+00:00"Dekimpe, Karel"https://www.zbmath.org/authors/?q=ai:dekimpe.karel"Tertooy, Sam"https://www.zbmath.org/authors/?q=ai:tertooy.sam"Vargas, Antonio R."https://www.zbmath.org/authors/?q=ai:vargas.antonio-rLet \(G\) be a group. The Reidemeister spectrum of \(G\) is the set
\[\text{Spec}_{R}(G) = \{R(\varphi)\mid \varphi \in \text{Aut}(G) \}, \]
where \(R(\varphi)\) is the Reidmeister number of \(\varphi.\)
In this paper, based on some computations of Reidemeister spectra, the authors prove that for any nonnegative integer \(n\) there exists a self-diffeomorphism \(h_n\) of \(M\) such that \(h_n\) has exactly \(n\) fixed points and any self-map \(f\) of \(M\) which is homotopic to \(h_n\) has at least \(n\) fixed points. Here, \(M\) is a nilmanifold with a fundamental group which is free 2-step nilpotent on at least 4 generators.
Reviewer: Weslem Liberato Silva (Ilhéus)Random groups, random graphs and eigenvalues of \(p\)-Laplacians.https://www.zbmath.org/1452.200372021-02-12T15:23:00+00:00"Druţu, Cornelia"https://www.zbmath.org/authors/?q=ai:drutu.cornelia"Mackay, John M."https://www.zbmath.org/authors/?q=ai:mackay.john-mSummary: We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on \(L^p\)-spaces (affine isometric, and more generally \((2 - 2 \varepsilon)^{1 / 2 p}\)-uniformly Lipschitz) with \(p\) varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal \(p\) for which \(L^p\)-fixed point properties hold and the conformal dimension of the boundary.
In the Gromov density model, we prove that for every \(p_0 \in [2, \infty)\) for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on \(L^p\)-spaces that are \((2 - 2 \varepsilon)^{1 / 2 p}\)-uniformly Lipschitz, and this for every \(p \in [2, p_0]\). To accomplish these goals we find new bounds on the first eigenvalue of the \(p\)-Laplacian on random graphs, using methods adapted from Kahn and Szemerédi's approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to \(L^p\)-spaces previous results for Kazhdan's Property (T) established by \textit{A. Żuk} [C. R. Acad. Sci., Paris, Sér. I 323, No. 5, 453--458 (1996; Zbl 0858.22007); Geom. Funct. Anal. 13, No. 3, 643--670 (2003; Zbl 1036.22004)] and \textit{W. Ballmann} and \textit{J. Świątkowski} [Geom. Funct. Anal. 7, No. 4, 615--645 (1997; Zbl 0897.22007)]i.On \(\mathbb{P} M\)-monoids and braid \(\mathbb{P} M\)-monoids.https://www.zbmath.org/1452.200612021-02-12T15:23:00+00:00"Miyatani, Toshinori"https://www.zbmath.org/authors/?q=ai:miyatani.toshinoriThe author defines two monoids analogous to the symmetric group \(S_n\) and the braid group \(B_n\), which are called \(\mathbb{P}M\) monoid and braid \(\mathbb{P}M\) monoid respectively. He shows that a \(\mathbb{P}M\) monoid has a presentation with generators and relators. Moreover he gives a solution to the word problem of the braid \(\mathbb{P}M\) monoid.
Reviewer: Stephan Rosebrock (Karlsruhe)Commensurations of subgroups of \(\operatorname{Out}(F_N)\).https://www.zbmath.org/1452.200302021-02-12T15:23:00+00:00"Horbez, Camille"https://www.zbmath.org/authors/?q=ai:horbez.camille"Wade, Richard D."https://www.zbmath.org/authors/?q=ai:wade.richard-dIn this paper, the authors consider the abstract and the relative commensurator groups of the outer automorphism groups of free groups. To begin with, we recall the definitions of the comensurator groups.
Let \(G\) be a group. Consider the equivalence relation on the set of isomorphisms \(\varphi: H \rightarrow N\) between finite index subgroups \(H\), \(N\) of \(G\) generated by the relation that \(\varphi_1: H_1 \rightarrow N_1\) is equivalent to \(\varphi_2 : H_2 \rightarrow N_2\) if \(\varphi_1 = \varphi_2\) on some finite index subgroup of \(G\). The abstract commensurator group \(\mathrm{Comm}(G)\) of \(G\) is the group of all equivalence classes of isomorphisms between finite index subgroups of \(G\) with respect to the equvalence relation. There is a natural homomorphism \(\mathrm{Aut}(G) \rightarrow \mathrm{Comm}(G)\). It is known that if \(G\) has the unique root property, \(\mathrm{Aut}(G)\) injects into \(\mathrm{Comm}(G)\). In genral, however, \(\mathrm{Comm}(G)\) is much larger than \(\mathrm{Aut}(G)\). For example, \(\mathrm{GL}(n,\mathbb Z) \cong \mathrm{Aut}(\mathbb Z^n) \subset \mathrm{Comm}(\mathbb Z^n) \cong \mathrm{GL}(n,\mathbb Q)\).
Two subgroups \(K_1\) and \(K_2\) of \(G\) are commensurable in \(G\) if \(K_1 \cap K_2\) are of finite index in both \(K_1\) and \(K_2\). Let \(\Gamma\) be a subgroup of \(G\). The relative commensurator group \(\mathrm{Comm}_G(\Gamma)\) of \(\Gamma\) in \(G\) is the subgroup of \(G\) consisting of all elements \(g \in G\) such that \(\Gamma\) and \(g \Gamma g^{-1}\) are commensurable in \(G\). There is a natural homomorphism \(\mathrm{Comm}_G(\Gamma) \rightarrow \mathrm{Comm}(\Gamma)\) induced by conjugation of \(G\).
The commensurbility is closely related to the study of a certain rigidity of groups. Today, it appears in several contexts in geometry and topology, including the studies of knots, \(3\)-dimensional hyperbolic manifolds, and so on. Let \(\mathcal{M}_g\) be the mapping class group of a closed oriented surface \(\Sigma_g\) of genus \(g\), and \(\mathcal{M}_g^{\pm}\) the extended mapping class group of \(\Sigma_g\). \textit{N. V. Ivanov} [Int. Math. Res. Not. 1997, No. 14, 651--666 (1997; Zbl 0890.57018)] showed that the natural map \(\mathcal{M}_g^{\pm} \rightarrow \mathrm{Comm}(\mathcal{M}_g)\) induced by the conjugation is an isomorphism by using the curve complex.
Let \(F_n\) be a free group of rank \(n\). \textit{B. Farb} and \textit{M. Handel} [Publ. Math., Inst. Hautes Étud. Sci. 105, 1--48 (2007; Zbl 1137.20018)] proved that the natural map \(\mathrm{Out}(F_n) \rightarrow \mathrm{Comm}(\mathrm{Out}(F_n))\) is an isomorphism for every \(n \geq 4\). Let \(\mathcal{H}_g\) be the handlebody group of genus \(g\). \textit{S. Hensel} [Comment. Math. Helv. 93, No. 2, 335--358 (2018; Zbl 1407.57003)] showed that \(\mathrm{Comm}(\mathcal{H}_g)\) is isomorphic to \(\mathcal{H}_g\) by studying rigidity of \(\mathcal{H}_g\) in \(\mathcal{M}_g\).
In this paper, the authors reprove the result of Farb and Handel in a different way, and generalize it to the case where \(n=3\). Based on a spirit of the method of Ivanov as mentioned above, the authors consider a certain simplicial \(\mathrm{Out}\,F_n\)-complex, called the edgewise nonseparating free splitting graph, denoted by \(\mathrm{FS}^{ens}\),
and study a rigidity of it. More precisely, \(\mathrm{FS}^{ens}\) is a graph whose vertices are the (homoeomorphism classes of) loop-edge free splittings of \(F_n\), and two vertices being joined by an edge whenever there are compatible and have a two-petal rose refinement. The authors show that the simplicial automorphisms of \(\mathrm{FS}^{ens}\)
all come from the \(\mathrm{Out}\,F_n\)-action.
The comensurbility of certain interesting subgroups of the mapping class groups has been studied so far.
\textit{B. Farb} and \textit{N. V. Ivanov} [``Torelli buildings and their automorphisms'', Preprint, \url{arXiv:1410.6223}] showed that the commensurator of the Toreli group is isomorphic to \(\mathcal{M}_g^{\pm}\). This fact is generalised to the kernel of the first Johnson homomorphism of \(\mathcal{M}_g\) by independent works of \textit{T. E. Brendle} and \textit{D. Margalit} [Geom. Topol. 8, 1361--1384 (2004; Zbl 1079.57017)], and \textit{Y. Kida} [Osaka J. Math. 50, No. 2, 309--337 (2013; Zbl 1282.20033)]. In this paper, the authors study the commensurators of certain subgroups of \(\mathrm{Out}\,F_n\), and show the followings.
Theorem. For \(n \geq 4\), let \(\Gamma\) be either
\begin{itemize}
\item a subgroup of \(\mathrm{Out}\,F_n\) which contains a term of the Andreadakis-Johnson filtration of \(\mathrm{Out}\,F_n\), or
\item a subgroup of \(\mathrm{Out}\,F_n\) that contains a power of every twist.
\end{itemize}
Then the natural map \(\mathrm{Comm}_{\mathrm{Out}\,F_n}(\Gamma) \rightarrow \mathrm{Comm}(\Gamma)\) is an isomorphism.
Theorem. Let \(\Gamma\) be either \(\mathrm{IA}_3\) or a subgroup of \(\mathrm{Out}\,F_3\) that contains a power of every twist. Then the natural map \(\mathrm{Comm}_{\mathrm{Out}\,F_3}(\Gamma) \rightarrow \mathrm{Comm}(\Gamma)\) is an isomorphism.
Reviewer: Takao Satoh (Tokyo)Pronormality of Hall subgroups in their normal closure.https://www.zbmath.org/1452.200122021-02-12T15:23:00+00:00"Vdovin, E. P."https://www.zbmath.org/authors/?q=ai:vdovin.evgeny-petrovitch"Nesterov, M. N."https://www.zbmath.org/authors/?q=ai:nesterov.m-n"Revin, D. O."https://www.zbmath.org/authors/?q=ai:revin.danila-olegovitchSummary: It is known that for any set \(\pi\) of prime numbers, the following assertions are equivalent: (1) in any finite group, \(\pi\)-Hall subgroups are conjugate; (2) in any finite group, \(\pi\)-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, \(\pi\)-Hall subgroups are pronormal in their normal closure. Previously [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka notebook. Unsolved problems in group theory. 18th edition. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2014; Zbl 1372.20001), Quest. 18.32], the question was posed whether it is true that in a finite group, \(\pi\)-Hall subgroups are always pronormal in their normal closure. Recently, the second author [Sib. Math. J. 58, No. 1, 128--133 (2017; Zbl 1367.20015); translation from Sib. Mat. Zh. 58, No. 1, 165--173 (2017)] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set \(\pi\). The fact that there exist examples of finite sets \(\pi\) and finite groups \(G\) such that \(G\) contains more than one conjugacy class of \(\pi\)-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for \(\pi\) is unessential for (1), (2), and (3) to be equivalent.Vanishing of \(\ell^2\)-Betti numbers of locally compact groups as an invariant of coarse equivalence.https://www.zbmath.org/1452.220012021-02-12T15:23:00+00:00"Sauer, Roman"https://www.zbmath.org/authors/?q=ai:sauer.roman"Schrödl, Michael"https://www.zbmath.org/authors/?q=ai:schrodl.michaelSummary: We provide a proof that the vanishing of \(\ell^2\)-Betti numbers of unimodular locally compact second countable groups is an invariant of coarse equivalence. To this end, we define coarse \(\ell^2\)-cohomology for locally compact groups and show that it is isomorphic to continuous cohomology for unimodular groups and invariant under coarse equivalence.A property of the lamplighter group.https://www.zbmath.org/1452.200222021-02-12T15:23:00+00:00"Castellano, I."https://www.zbmath.org/authors/?q=ai:castellano.ilaria"Cook, G. Corob"https://www.zbmath.org/authors/?q=ai:corob-cook.ged"Kropholler, P. H."https://www.zbmath.org/authors/?q=ai:kropholler.peter-hA subgroup \(H\) of a group \(G\) is said to be inert (commensurated in another terminology) if \(H\) and \(H^g=g^{-1}Hg\) are commensurate for all \(g\in G\), meaning that \(H\cap H^g\) always has a finite index in both \(H\) and \(H^g\).
It is shown that the inert subgroups of the lamplighter group \(\mathbb{Z}_pwr\mathbb{Z}, p \) is a prime, fall into exactly five commensurability classes.
An application of this result to the theory of totally disconnected locally compact groups is given. A dynamical aspect of the property investigated in the paper related to the concept of algebraic entropy is also presented.
Reviewer: V. A. Roman'kov (Omsk)A generalization to Sylow permutability of pronormal subgroups of finite groups.https://www.zbmath.org/1452.200102021-02-12T15:23:00+00:00"Esteban-Romero, R."https://www.zbmath.org/authors/?q=ai:esteban-romero.ramon"Longobardi, P."https://www.zbmath.org/authors/?q=ai:longobardi.patrizia"Maj, M."https://www.zbmath.org/authors/?q=ai:maj.mercedeThe notion of \(T\)-groups (groups in which normality is a transitive relation) was introduced by [\textit{E. Best} and \textit{O. Taussky}, Proc. R. Ir. Acad., Sect. A 47, 55--62 (1942; Zbl 0063.00359)] and also the first investigations were done. Lateron (see [\textit{G. Zacher}, Ric. Mat. 1, 287--294 (1952; Zbl 0081.25701)]) and \textit{W. Gaschütz} [J. Reine Angew. Math. 198, 87--92 (1957; Zbl 0077.25003)] obtained classifications of \(T\)-groups in the solvable universe. The classification of \(T\) this paper refers to is due to [\textit{G. Kaplan}, Arch. Math. 96, No. 1, 19--25 (2011; Zbl 1230.20020)] and [\textit{T. A. Peng}, Proc. Am. Math. Soc. 20, 232--234 (1969; Zbl 0167.02302)]. A solvable group is a \(T\)-group iff all \(p\)-subgroups (for every prime \(p\)) are pronormal iff all subgroups are pronormal. The authors study \(PT\)-groups (permutability is a transitive relation ) and PST-groups (permutability with Sylow groups is a transitive relation). They show that a solvable group is a \(PT\)-group if all subgroups are pro permutable iff all \(p\)-subgroups are pro permutable. Here, a group \(H\) is called pro permutable if for all \(g \in G\) there is some \(x\) in the nilpotent residue of \(\langle H, H^g\rangle\) such that \(H\) is permutable in \(\langle H,gx^{-1}\rangle\). There is also a corresponding classification of solvable \(PST\)-groups where one just replaces permutable by \(S\)-permutable everywhere.
Reviewer: Gernot Stroth (Halle)Euclidean-valued group cohomology is always reduced.https://www.zbmath.org/1452.200512021-02-12T15:23:00+00:00"Austin, Tim"https://www.zbmath.org/authors/?q=ai:austin.tim-dRealizable ranks of joins and intersections of subgroups in free groups.https://www.zbmath.org/1452.200192021-02-12T15:23:00+00:00"Soroko, Ignat"https://www.zbmath.org/authors/?q=ai:soroko.ignatAuthor's abstract: The famous Hanna Neumann Conjecture (now the Friedman-Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups \(H\) and \(K\) of a non-abelian free group. It is an interesting question to ``quantify'' this bound with respect to the rank of \(H \vee K\), the subgroup generated by \(H\) and \(K\). We describe a set of realizable values \((rk(H \vee K), rk(H \cap K))\) for arbitrary \(H\), \(K\), and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for \(H\) and \(K\) with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of \( H \vee K\), \(H \cap K\) are not realizable, thus resolving the remaining open case \(m = 4\) of Guzman's ``Group-Theoretic Conjecture'' in the affirmative. This in turn implies the validity of the corresponding ``Geometric Conjecture'' on hyperbolic \(3\)-manifolds with a \(6\)-free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when \(rk(H) = 2\).
Reviewer: Alexander Ivanovich Budkin (Barnaul)Vertex operator algebras, number theory and related topics. International conference, California State University, Sacramento, CA, USA, June 11--15, 2018.https://www.zbmath.org/1452.170022021-02-12T15:23:00+00:00"Krauel, Matthew (ed.)"https://www.zbmath.org/authors/?q=ai:krauel.matthew"Tuite, Michael (ed.)"https://www.zbmath.org/authors/?q=ai:tuite.michael-p"Yamskulna, Gaywalee (ed.)"https://www.zbmath.org/authors/?q=ai:yamskulna.gaywaleePublisher's description: This volume contains the proceedings of the International Conference on Vertex Operator Algebras, Number Theory, and Related Topics, held from June 11--15, 2018, at California State University, Sacramento, California.
The mathematics of vertex operator algebras, vector-valued modular forms and finite group theory continues to provide a rich and vibrant landscape in mathematics and physics. The resurgence of moonshine related to the Mathieu group and other groups, the increasing role of algebraic geometry and the development of irrational vertex operator algebras are just a few of the exciting and active areas at present.
The proceedings center around active research on vertex operator algebras and vector-valued modular forms and offer original contributions to the areas of vertex algebras and number theory, surveys on some of the most important topics relevant to these fields, introductions to new fields related to these and open problems from some of the leaders in these areas.
The articles of this volume will be reviewed individually.On normal subgroups of the unit group of a quaternion algebra over a Pythagorean field.https://www.zbmath.org/1452.160362021-02-12T15:23:00+00:00"Mahmoudi, Mohammad Gholamzadeh"https://www.zbmath.org/authors/?q=ai:mahmoudi.mohammad-gholamzadehSummary: We investigate the structure of normal subgroups of the unit group of a quaternion algebra over a Pythagorean field.Engel elements in weakly branch groups.https://www.zbmath.org/1452.200342021-02-12T15:23:00+00:00"Fernández-Alcober, Gustavo A."https://www.zbmath.org/authors/?q=ai:fernandez-alcober.gustavo-a"Noce, Marialaura"https://www.zbmath.org/authors/?q=ai:noce.marialaura"Tracey, Gareth M."https://www.zbmath.org/authors/?q=ai:tracey.gareth-mA \textit{branch group} is a special kind of group acting on spherically homogeneous rooted trees.
This paper is devoted to the study of properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, it is proved that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all \(p\)-elements and that the group is virtually a \(p\)-group for some prime \(p\). It is also showed that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition.
Reviewer: Egle Bettio (Venezia)The quantum group \(S L_q^{\star}(2)\) and quantum algebra \(U_q(s l_2^{\star})\) based on a new associative multiplication on \(2 \times 2\) matrices.https://www.zbmath.org/1452.811312021-02-12T15:23:00+00:00"Aziziheris, K."https://www.zbmath.org/authors/?q=ai:aziziheris.kamal"Fakhri, H."https://www.zbmath.org/authors/?q=ai:fakhri.hossein"Laheghi, S."https://www.zbmath.org/authors/?q=ai:laheghi.samanehSummary: We present classical groups \(SL^\star (2)\) and \(SU^\star (2)\) as well as classical Lie algebra \(s l_2^\star(\mathbb{C})\) associated with a new associative multiplication on \(2 \times 2\) matrices. The idea of the new multiplication is generalized to the action of a \(2 \times 2\) square matrix on a \(2 \times 1\) column one. The coordinate Hopf algebra \(\mathcal{O}(S L_q^\star(2))\) is introduced as a \(q\)-generalization of Hopf algebra \(\mathcal{O}(S L^\star(2))\), and it is shown that the coordinate algebra corresponding to the quantum plane \(\mathbb{C}_q^2\) is a \(S L_q^\star(2)\)-left-covariant algebra. Furthermore, the quantized universal enveloping algebra \(U_q(s l_2^\star)\) with the Hopf structure as a dual of \(\mathcal{O}(S L_q^\star(2))\) is introduced. For each of the Hopf algebras \(\mathcal{O}(S L_q^\star(2))\) and \(U_q(s l_2^\star)\), we associate two different real forms with two inequivalent families of *-involutions, with \(\mathcal{O}(S U_q^\star(2))\) and \(U_q(s u_2^\star)\) as one of the real forms. It is shown that the Hopf algebra pairing is a dual pairing of two Hopf *-algebras \(\mathcal{O}(S U_q^\star(2))\) and \(U_q(s u_2^\star)\).
{\copyright 2020 American Institute of Physics}Embedding the set of nondivisorial ideals of a numerical semigroup into \(\mathbb N^n\).https://www.zbmath.org/1452.200572021-02-12T15:23:00+00:00"Spirito, Dario"https://www.zbmath.org/authors/?q=ai:spirito.darioProperties of annihilator graph of a commutative semigroup.https://www.zbmath.org/1452.050852021-02-12T15:23:00+00:00"Talebi, Yahya"https://www.zbmath.org/authors/?q=ai:talebi.yahya"Akbarzadeh, Sahar"https://www.zbmath.org/authors/?q=ai:akbarzadeh.saharSummary: Let \(S\) be a commutative semigroup with zero. Let \(Z(S)\) be the set of all zero-divisors of \(S\). We define the annihilator graph of \(S\), denoted by \(\operatorname{ANN}_G(S)\), as the undirected graph whose set of vertices is \(Z(S)^{\ast}=Z(S)-\{0\}\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(\operatorname{ann}_S(xy)\neq \operatorname{ann}_S(x)\cap \operatorname{ann}_S(y)\). In this paper, we study some basic properties of \(\operatorname{ANN}_G(S)\) by means of \(\Gamma(S)\). We also show that if \(Z(S)\neq S\), then \(\operatorname{ANN}_G(S)\) is a subgraph of \(\Gamma(S)\). Moreover, we investigate some properties of the annihilator graph \(\operatorname{ANN}_G(S)\) related to minimal prime ideals of \(S\). We also study some connections between the domination numbers of annihilator graphs and zero-divisor graphs.Skew Pieri algebras of the general linear group.https://www.zbmath.org/1452.200442021-02-12T15:23:00+00:00"Kim, Sangjib"https://www.zbmath.org/authors/?q=ai:kim.sangjib"Lee, Soo Teck"https://www.zbmath.org/authors/?q=ai:lee.soo-teck"Wang, Yi"https://www.zbmath.org/authors/?q=ai:wang.yi.6|wang.yi.3|wang.yi.1|wang.yi.10|wang.yi.5|wang.yi.7|wang.yi.8|wang.yi.9|wang.yi.4|wang.yi.2Summary: Let \(V\) be an irreducible polynomial representation of the general linear group \(\mathrm{GL}_n = \mathrm{GL}_n(\mathbb{C})\) and let \(\alpha_1, \dots,\alpha_q\) be nonnegative integers less than or equal to \(n\). We call a description of the irreducible decomposition of the tensor product \(V \otimes \operatorname{\Lambda}^{\alpha_1}(\mathbb{C}^n) \otimes \cdots \otimes \operatorname{\Lambda}^{\alpha_q}(\mathbb{C}^n)\) an \textit{iterated skew Pieri rule} for \(\mathrm{GL}_n\). In this paper, we define a family of complex algebras whose structure encodes an iterated skew Pieri rule for \(\mathrm{GL}_n\), and we call these algebras \textit{iterated skew Pieri algebras.} Our main goal is to construct a basis for each of these algebras thereby giving explicit highest weight vectors in the above tensor product.{
\copyright 2018 American Institute of Physics}Classification of Coxeter groups with finitely many elements of \(\mathfrak{a}\)-value 2.https://www.zbmath.org/1452.200352021-02-12T15:23:00+00:00"Green, R. M."https://www.zbmath.org/authors/?q=ai:green.richard-m"Xu, Tianyuan"https://www.zbmath.org/authors/?q=ai:xu.tianyuanThis paper concerns the Kazhdan-Lusztig basis of a Coxeter group \(W\), and the associated \(\mathbf a\)-function \(W\to\mathbb N\). This function is defined in terms of the structure constants for the Kazhdan-Lusztig basis, and is important in understanding the Kazhdan-Lusztig cells of \(W\); in particular, the function is constant on each cell.
The aim of the paper is a straightforward one: to determine which Coxeter groups are \textit{\(\mathbf a(2)\)-finite}, i.e. have only finitely many elements \(w\) with \(\mathbf a(w)=2\). This means that in some sense the Coxeter group is ``not too infinite'', and the main theorem gives a complete classification of \(\mathbf a(2)\)-finite groups in terms of Coxeter graphs. Interestingly, the classification includes some, but not all, Coxeter groups of affine type.
The proof involves a variety of tools, mostly managing to avoid explicit work with the Kazhdan-Lusztig basis: some known results on \(\mathbf a\)-values (in particular, those relating to the ``star operations''), and study of heaps associated to fully commutative elements of Coxeter groups. The paper is very well written, giving an excellent introduction to the subject and the tools used, and presenting the proof with a suitable level of detail.
Reviewer: Matthew Fayers (London)Some properties of left-transitive quasigroups.https://www.zbmath.org/1452.200622021-02-12T15:23:00+00:00"Didurik, N. N."https://www.zbmath.org/authors/?q=ai:didurik.natalia-nSummary: Properties of left-transitive quasigroups (including their autotopisms and pseudoautomorphisms), their connections with some quasigroup classes are established. Left-transitive right GA-quasigroups are described.Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category.https://www.zbmath.org/1452.170122021-02-12T15:23:00+00:00"Hu, Jun"https://www.zbmath.org/authors/?q=ai:hu.jun.1"Lam, Ngau"https://www.zbmath.org/authors/?q=ai:lam.ngauLet \(\mathfrak{g}\) be a complex semisimple Lie algebra with a fixed Borel subalgebra \(\mathfrak{b}\) and let \(\mathfrak{p}\) be a parabolic subalgebra containing \(\mathfrak{b}\). Let \(\mathcal{O}^{\mathfrak{p}}\) denote the corresponding parabolic Bernstein-Gelfand-Gelfand (BGG) category. For each dominant integral weight \(\lambda\), let \(\mathcal{O}^{\mathfrak{p}}_{\lambda}\) denote the subcategory of \(\mathcal{O}^{\mathfrak{p}}\) consisting of modules whose composition factors have highest weights of the form \(w \cdot \lambda\), where \(w\) is an element of the Weyl group \(W\) of \(\mathfrak{g}\) and \(\cdot\) denotes the dot action of \(W\). In the paper under review, the authors consider a conjecture of Khovanov, which states that for any dominant integral weight \(\lambda\), the endomorphism algebra of each projective-injective module (i.e., at the same time projective and injective) in \(\mathcal{O}^{\mathfrak{p}}_{\lambda}\) is a symmetric algebra. When \(\lambda\) is a regular weight, the conjecture is proven in [\textit{V. Mazorchuk} and \textit{C. Stroppel}, J. Reine Angew. Math. 616, 131--165 (2008; Zbl 1235.16013)]. The main purpose of the paper under review is to give a proof of Khovanov's conjecture for the case of a singular \(\lambda\) (Theorem 1.2).
In Section 2, the authors define the basic projective-injective module over any finite-dimensional algebra \(A\) and study the endomorphism algebra \(B\) of this basic projective-injective module in a general setting. Given an appropriate basis for \(B\), the authors define a canonical form ``tr'' attached to it (and consequently an induced associative bilinear form \((-, -)_{\mathrm{tr}}\)). They show that if the algebra \(A\) is equipped with an anti-involution fixing each simple \(A\)-module, then the bilinear form \((-, -)_{\mathrm{tr}}\) on \(B\) is non-degenerate. Therefore, \(B\) is a Frobenius algebra. The authors also give necessary conditions for \(B\) to be symmetric.
Then, in Section 3, the authors of the paper under review consider the case when \(A\) is a \(\mathbb{Z}\)-graded finite-dimensional algebra. They define the notion of an \textit{admissible basis} for \(B\) (Definition 3.7) and show that if an admissible basis for \(B\) exists, then the canonical form ``tr'' attached to it is always symmetric and hence \(B\) is a symmetric algebra. Furthermore, for certain positively \(\mathbb{Z}\)-graded finite-dimensional algebras \(A\), \(B\) is a symmetric algebra if and only if there exists an admissible basis for \(B\).
In Section 4, using the results from Sections 2 and 3, the authors give a proof of Theorem 1.2. Finally, in the Appendix, a short proof of Theorem 1.2 due to Coulembier and Mazorchuk is given.
Reviewer: Elitza Hristova (Sofia)Lifting involutions in a Weyl group to the torus normalizer.https://www.zbmath.org/1452.200452021-02-12T15:23:00+00:00"Lusztig, G."https://www.zbmath.org/authors/?q=ai:lusztig.georgeSummary: Let \( N\) be the normalizer of a maximal torus \( T\) in a split reductive group over \( F_q\), and let \( w\) be an involution in the Weyl group \( N/T\). We explicitly construct a lifting \( n\) of \( w\) in \( N\) such that the image of \( n\) under the Frobenius map is equal to the inverse of \( n\).Finite 2-groups with a non-Dedekind non-metacyclic norm of abelian non-cyclic subgroups.https://www.zbmath.org/1452.200132021-02-12T15:23:00+00:00"Lyman, Fedir"https://www.zbmath.org/authors/?q=ai:lyman.fedir-m"Lukashova, Tetyana"https://www.zbmath.org/authors/?q=ai:lukashova.tetyana-d"Drushlyak, Marina"https://www.zbmath.org/authors/?q=ai:drushlyak.marinaSummary: The authors study finite 2-groups with non-Dedekind non-metacyclic norm \(N_G^A\) of abelian non-cyclic subgroups depending on the cyclicness or the non-cyclicness of the center of a group \(G\). The norm \(N_G^A\) is defined as the intersection of the normalizers of abelian non-cyclic subgroups of \(G\). It is found out that such 2-groups are cyclic extensions of their norms of abelian non-cyclic subgroups. Their structure is described.A Bieberbach theorem for crystallographic group extensions.https://www.zbmath.org/1452.200502021-02-12T15:23:00+00:00"Ratcliffe, John G."https://www.zbmath.org/authors/?q=ai:ratcliffe.john-g"Tschantz, Steven T."https://www.zbmath.org/authors/?q=ai:tschantz.steven-tSummary: In this paper, we prove that for each dimension \(n\), there are only finitely many isomorphism classes of pairs of groups \((\Gamma,N)\) such that \(\Gamma\) is an \(n\)-dimensional crystallographic group and \(N\) is a normal subgroup of \(\Gamma\) such that \(\Gamma/N\) is a crystallographic group. This result is equivalent to the statement that for each dimension \(n\) there are only finitely many affine equivalence classes of geometric orbifold fibrations of compact, connected and flat \(n\)-orbifolds.Essential surfaces in graph pairs.https://www.zbmath.org/1452.200382021-02-12T15:23:00+00:00"Wilton, Henry"https://www.zbmath.org/authors/?q=ai:wilton.henrySummary: A well-known question of Gromov asks whether every one-ended hyperbolic group \( \Gamma \) has a surface subgroup. We give a positive answer when \( \Gamma \) is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov's question is reduced (modulo a technical assumption on 2-torsion) to the case when \( \Gamma \) is rigid. We also find surface subgroups in limit groups. It follows that a limit group with the same profinite completion as a free group must in fact be free, which answers a question of Remeslennikov in this case.A natural partial order on certain semigroups of transformations restricted by an equivalence.https://www.zbmath.org/1452.200592021-02-12T15:23:00+00:00"Han, X."https://www.zbmath.org/authors/?q=ai:han.xiaosen|han.xiaozhen|han.xueshuo|han.xiyue|han.xinyu|han.xingbo|han.xuehong|han.xinxing|han.xiaoyi|han.xinzhong|han.xiaoyou|han.xiangli|han.xuliang|han.xinhuan|han.xuchen|han.xiaowei|han.xiaofang|han.xianli|han.xiaole|han.xiaojing|han.xiangping|han.xiaopu|han.xiaoran|han.xiaodong|han.xianglan|han.xueliang|han.xuejing|han.xiaocong|han.xuanli|han.xun|han.xie|han.xiannan|han.xiaoning|han.xingming|han.xiuyou|han.xiaolin|han.xucang|han.xiaoli|han.xiu|han.xiujing|han.xiaoyu|han.xingling|han.xuefei|han.xianlin|han.xiaoshuang|han.xinghui|han.xaoling|han.xiaoqin|han.xuli|han.xinxin|han.xiaofeng|han.xiangling|han.xuesong|han.xianhua|han.xinfang|han.xiaomin|han.xiaoxin|han.xue|han.xintian|han.xiao|han.xiaoliang|han.xiaolei|han.xiaoxi|han.xiangyan|han.xiaosong|han.xiaojuan|han.xiaohong|han.xiaoxu|han.xiaoling|han.xinrui|han.xiangxi|han.xingbao|han.xia|han.xinglin|han.xiaoru|han.xi-an|han.xinli|han.xinwei|han.xiaohui|han.xixuan|han.xingyu|han.xiaoyun|han.xiwu|han.xueli|han.xiaoying|han.xizhi|han.xuexian|han.xiaofei|han.xixian|han.xuefeng|han.xudong|han.xin|han.xinping|han.xu|han.xiaoming|han.xiaoxia|han.xixi|han.xiurong|han.xiaojun|han.xianglin|han.xiang|han.xuan|han.xiangke|han.xiaohu|han.xiaotian|han.xiangzhu|han.xiaoyan|han.xiaozhuo|han.xiaopan|han.xiaomei|han.xiaoya|han.xiaoguang|han.xinqiang|han.xingsi|han.xiuping|han.xiaobeng|han.xianhong|han.xiaohua|han.xianjun|han.xiying|han.xinglou|han.xianquan|han.xinyin|han.xuemei|han.xuefang|han.xueping|han.xiaoxue|han.xiaori"Sun, L."https://www.zbmath.org/authors/?q=ai:sun.luoyi|sun.linxuan|sun.lihuan|sun.longsheng|sun.lucheng|sun.liling|sun.linzhi|sun.luping|sun.leyuan|sun.linyan|sun.lishuang|sun.lianlong|sun.laijun|sun.luning|sun.liangxin|sun.lianshan|sun.longfa|sun.longjie|sun.lingzhou|sun.lulu|sun.linan|sun.linpo|sun.linchun|sun.lijun|sun.liqin|sun.lisheng|sun.lianyou|sun.linman|sun.lingiang|sun.liang|sun.lihua|sun.libin|sun.lizhi|sun.lingyu|sun.lina|sun.luming|sun.lingling|sun.long|sun.liangjie|sun.ling|sun.limei|sun.lingjiao|sun.lingli|sun.liuquan|sun.libo|sun.liqun|sun.liuping|sun.lingliang|sun.litao|sun.lixia|sun.liankun|sun.lixiang|sun.lunkai|sun.liuying|sun.lijing|sun.lixin|sun.lijie|sun.longxiang|sun.lanxiang|sun.linfa|sun.liquan|sun.linxin|sun.lingfang|sun.lianju|sun.leiyu|sun.limin|sun.lujie|sun.liyuan|sun.lifeng|sun.linping|sun.licheng|sun.lidan|sun.lu|sun.liangji|sun.liping|sun.liming|sun.lizhu|sun.ligang|sun.liangliang|sun.lirong|sun.liqiang|sun.le|sun.lily|sun.liangrui|sun.lishan|sun.liangping|sun.lining|sun.leping|sun.liya|sun.lichiang|sun.lei|sun.lili|sun.lianzhi|sun.lin-hui|sun.liuguan|sun.lianming|sun.linlin|sun.liwei|sun.lin|sun.lanyin|sun.lixing|sun.linjie|sun.liuxin|sun.lijuan|sun.liyan|sun.libao|sun.litan|sun.luna|sun.lelin|sun.li|sun.liewu|sun.liangtian|sun.lan|sun.lihui|sun.lihong|sun.lijian|sun.laixiang|sun.lanfen|sun.liying|sun.libing|sun.longgangSummary: Let \(\sigma \) be an equivalence on \(X\) and let \(E(X,\sigma)\) denote the semigroup (under composition) of all \(f:X\rightarrow X\), such that \(\sigma \subseteq \operatorname{ker}(f)\). In this paper, we endow the semigroup \(E(X,\sigma)\) with a well-known natural partial order \(\leq \) and provide a characterization for \(\leq \) and prove necessary and sufficient conditions for \(\leq \) to be both left and right compatible with the multiplication. We also describe the minimal and the maximal elements of \(E(X,\sigma)\) with respect to this order.Locally \(E\)-solid epigroups.https://www.zbmath.org/1452.200562021-02-12T15:23:00+00:00"Liu, J. G."https://www.zbmath.org/authors/?q=ai:liu.jiangen|liu.jingge|liu.jingai|liu.jiaguo|liu.jian-guo.1|liu.jigang|liu.jingbo|liu.jingou|liu.jingang|liu.jungang|liu.jingguo|liu.jiangang|liu.jiangguo|liu.jingun|liu.jiagang|liu.jiageng|liu.jian-guo|liu.j-gary|liu.jianguang|liu.jinguiSummary: An epigroup is a semigroup in which some power of any element lies in a subgroup of the given semigroup. The aim of the paper is to characterize epigroups which are locally \(E\)-solid in terms of ``forbidden'' epidivisors, in terms of certain decompositions as well as in terms of identities. As a subclass of locally \(E\)-solid epigroups, epigroups which are locally in semilattices of archimedean epigroups are also described from different points.On strongly primary monoids and domains.https://www.zbmath.org/1452.130022021-02-12T15:23:00+00:00"Geroldinger, Alfred"https://www.zbmath.org/authors/?q=ai:geroldinger.alfred"Roitman, Moshe"https://www.zbmath.org/authors/?q=ai:roitman.mosheThe authors study the local tameness of integral domains and strongly primary monoids. This study allows them to solve Problem 38 of [\textit{P.-J. Cahen} et al., in: Commutative algebra. Recent advances in commutative rings, integer-valued polynomials, and polynomial functions. Based on mini-courses and a conference on commutative rings, integer-valued polynomials and polynomial functions, Graz, Austria, December 16--18 and December 19--22, 2012. New York, NY: Springer. 353--375 (2014; Zbl 1327.13002)] which pose the question: are one-dimensional local Mori domains locally tame?
The paper contains several technical lemmas which are needed in order to prove the main results. The authors also give a characterization of strongly primary monoids and strongly primary domains with the property of being globally tame. These theorems give a positive answer to the question about the tameness of one-dimensional local Mori domains.
The paper provides the reader with all the basic background needed for being able to understand it. Good examples are shown to illustrate the results.
Reviewer: Daniel Marín Aragon (Cádiz)On sums of Sylow numbers of finite groups.https://www.zbmath.org/1452.200092021-02-12T15:23:00+00:00"Asboei, Alireza Khalili"https://www.zbmath.org/authors/?q=ai:khalili-asboei.alireza"Darafsheh, Mohammad Reza"https://www.zbmath.org/authors/?q=ai:darafsheh.mohammad-rezaSummary: Let \(G\) be a finite group. Let \(n_{p}(G)\) be the number of Sylow \(p\) subgroup of \(G\) and \(\pi (G)\) be the set of prime divisors of \(|G|\). We set \(S(G)=\{p\in \pi (G)\mid n_{p}(G)>1\}\) and \(\delta (G)=\sum_{p\in \pi (G)}n_{p}(G)\), and \( \delta_{0}(G)=\sum_{p\in S(G)}n_{p}(G)\). In this paper, we study groups \(G\) with small \(\delta (G)\) and \(\delta_{0}(G)\). Furthermore, we will show that if \(G\) is a non-solvable group with \(C_{G}(N)=\{1\}\), where the minimal normal subgroup \(N\) of \(G\) is the last member of the derived series of \(G\), then \(|G:G'|<\delta_{0}(G)\).On minimal generating sets for symmetric and alternating groups.https://www.zbmath.org/1452.200032021-02-12T15:23:00+00:00"Iradmusa, Moharram N."https://www.zbmath.org/authors/?q=ai:iradmusa.moharram-n"Taleb, Reza"https://www.zbmath.org/authors/?q=ai:taleb.rezaSummary: By a famous result, the subgroup generated by the \(n\)-cycle \(\sigma =(1,2,\dots,n)\) and the transposition \(\tau =(a,b)\) is the full symmetric group \(S_{n}\) if and only if \(\gcd(n,b-a)=1\). In this paper, we first generalize the above result for one \(n\)-cycle and \(k\) arbitrary transpositions, and then provide similar necessary and sufficient conditions for the subgroups of \(S_{n}\) in the following three cases: first, the subgroup generated by the \(n\)-cycle \(\sigma \) and a 3-cycle \(\delta =(a,b,c)\), second, the subgroup generated by the \(n\)-cycle \(\sigma \) and a set of transpositions and 3-cycles, and third, by the \(n\)-cycle \(\sigma \) and an involution \((a,b)(c,d)\). In the first case, we also determine the structure of the subgroup generated by \((1,2,\dots,n)\) and a 3-cycle \(\delta =(a,b,c)\) in general. Finally, an application to unsolvability of a certain infinite family of polynomials by radicals is given.On weakly \(\mathcal {M}\)-supplemented subgroups and the \(\mathcal {F}\)-hypercentre of finite groups.https://www.zbmath.org/1452.200162021-02-12T15:23:00+00:00"Miao, Liyun"https://www.zbmath.org/authors/?q=ai:miao.liyun"Guo, Xiuyun"https://www.zbmath.org/authors/?q=ai:guo.xiuyunSummary: A subgroup \(H\) of a finite group \(G\) is said to be weakly \(\mathcal {M}\)-supplemented in \(G\) if there exists a subgroup \(B\) of \(G\) such that (1) \(G=HB\) and (2) \(H_1B = BH_1 <G\) if \(H_1\)/\(H_G\) is a maximal subgroup of \(H/H_G\), where \(H_G\) is the largest normal subgroup of \(G\) contained in \(H\). In this paper, we investigate the structure of the \(\mathcal {F}\)-hypercentre \(Z_{\mathcal {F}}(G)\) in a finite group \(G\) using some family of weakly \(\mathcal {M}\)-supplemented subgroups in \(G\), where \(\mathcal {F}\) is a solubly saturated formation containing the class of all finite supersolvable groups.On fixed points of automorphisms.https://www.zbmath.org/1452.200272021-02-12T15:23:00+00:00"Atabekyan, V. S."https://www.zbmath.org/authors/?q=ai:atabekyan.varuzhan-s"Aslanyan, H. T."https://www.zbmath.org/authors/?q=ai:aslanyan.haika-tSummary: We prove that each automorphism of order 2 of any non-abelian periodic group of odd period has a non-trivial fixed point.On generalized modular subgroups of finite groups.https://www.zbmath.org/1452.200082021-02-12T15:23:00+00:00"Hu, B."https://www.zbmath.org/authors/?q=ai:hu.bingmin|hu.busong|hu.benqiong|hu.baozhu|hu.baiding|hu.bijin|hu.bixin|hu.baowen|hu.benyong|hu.baoqing|hu.boxing|hu.bowen|hu.bitao|hu.bei|hu.bing|hu.baogang|hu.bingzhong|hu.binbin|hu.biao|hu.beilai|hu.baomin|hu.bingsong|hu.brian|hu.binglu|hu.benmu|hu.bingran|hu.baocun|hu.baiqing|hu.bingquan|hu.bin|hu.bang|hu.bingyang|hu.beibei|hu.baosheng|hu.baohua|hu.beihua|hu.biaobiao|hu.bangyou|hu.botao|hu.bizhong|hu.baokun|hu.binjie|hu.bo|hu.baolin|hu.baoan|hu.boxia|hu.bambi|hu.binxin|hu.baoli"Huang, J."https://www.zbmath.org/authors/?q=ai:huang.jianwu|huang.jialin|huang.jiaxiang|huang.jiayuan|huang.jinhong|huang.jirong|huang.jieqing|huang.jia-yen|huang.jianke|huang.jianjun|huang.jun|huang.jyunping|huang.jingyu|huang.jizhou|huang.jinchao|huang.jingdong|huang.jiaqian|huang.jinzhi|huang.jinchun|huang.jiansong|huang.jiwu|huang.jianfei|huang.jiqi|huang.jiongtao|huang.jianmei|huang.jingjing|huang.jia|huang.jingxiong|huang.jieshan|huang.jiwei|huang.jijun|huang.jui-chi|huang.jiayin|huang.jin|huang.junqi|huang.jiangyang|huang.jianwen|huang.jiawei|huang.jiazhou|huang.jingfeng|huang.jizu|huang.junzhen|huang.jianmin|huang.jieying|huang.jinghao.1|huang.jiangqiang|huang.jianye|huang.jing-song|huang.junhui|huang.junzhou|huang.jinke|huang.junli|huang.jinjie|huang.jianping|huang.jianhui|huang.jingxi|huang.jiajian|huang.jianchao|huang.jiangping|huang.jizheng|huang.jiamin|huang.jingqi|huang.jialiang|huang.jinyang|huang.jiangshuai|huang.jianqing|huang.jingbo|huang.jiaji|huang.jinneng|huang.jiaxing|huang.junfei|huang.jiang|huang.jinshan|huang.jianyong|huang.junming|huang.jiaoyang|huang.jian.2|huang.jiarao|huang.jingjun|huang.jiajing|huang.jinming|huang.jiagui|huang.jiancai|huang.joshua|huang.jianning|huang.jihua|huang.jie.2|huang.jinrui|huang.jiabin|huang.juanjuan|huang.jingui|huang.jianguo|huang.jianbing|huang.jili|huang.jiandong|huang.jinli|huang.jiazhao|huang.jincheng|huang.jinquan|huang.junwen|huang.junwu|huang.jinshu|huang.jintang|huang.jiapeng|huang.jianwei|huang.jidan|huang.jiyan|huang.jinping|huang.jiewu|huang.jicheng|huang.jiaoru|huang.junheng|huang.jitang|huang.jinbo|huang.juan|huang.jida|huang.jianrong|huang.jingxiang|huang.jiangyan|huang.jinfeng|huang.jeffrey|huang.jianzhe|huang.jingyin|huang.jiangchuan|huang.ji|huang.jinxiang|huang.junliang|huang.junfu|huang.jianliang|huang.jinying|huang.jianbo|huang.jinggang|huang.junmin|huang.jinyuan|huang.jefferson|huang.jiping|huang.jinghua|huang.juntao|huang.jinguo|huang.jianxiong|huang.junjian|huang.jianfeng|huang.jianchun|huang.jici|huang.jiong|huang.jinjin|huang.jiegan|huang.jingchuan|huang.jingsi|huang.jinrong|huang.jingchi|huang.jinhua|huang.junjie|huang.jishan|huang.jiaming|huang.jingpin|huang.jinwu|huang.junhua|huang.jingchun|huang.jiacai|huang.jingli|huang.junbo|huang.jianming|huang.jiaqi|huang.jinlin|huang.jiumei|huang.jicai|huang.jigang|huang.jennifer|huang.jingwen|huang.jianxin|huang.jingshan|huang.jinwei|huang.jincai|huang.jonathan|huang.jianzhong|huang.jue|huang.junhong|huang.jiexiang|huang.jianzhi|huang.jianbin|huang.jiangyin|huang.juqing|huang.jianzhen|huang.jiaying|huang.jingang|huang.jie.1|huang.jiqing|huang.juyong|huang.jingfang|huang.jiahan|huang.jiuzhou|huang.jiayue|huang.jiawen|huang.jingcong|huang.jinxia|huang.jingquan|huang.jerry|huang.jiahao|huang.jinliang|huang.jingmin|huang.jiaxi|huang.jiuke|huang.jinjing|huang.jianqiang|huang.jiasheng|huang.jiangtao|huang.jianing|huang.junwei|huang.jiebin|huang.juiyu|huang.jieru|huang.jianbang|huang.jihong|huang.jixiang|huang.jifeng|huang.jiaqing|huang.jihai|huang.ju|huang.jinyu|huang.jianyu|huang.jiacheng|huang.jianhua|huang.jian|huang.jingwei|huang.jinsong|huang.jinlong|huang.jianyuan|huang.jianhong|huang.junqin|huang.jiting|huang.jianzhuang|huang.jiangke|huang.jen-tsung|huang.jing|huang.jianyi|huang.jincun|huang.jintong|huang.jianbai|huang.jinsheng|huang.jilin|huang.jianpeng|huang.jiahuiSummary: Let \(G\) be a finite group and \(M\) a subgroup of \(G\). Then \(M\) is called \textit{modular} if the following conditions are held: (1) \(\langle X, M \cap Z \rangle =\langle X, M \rangle \cap Z\) for all \(X \leq G, Z \leq G\) such that \(X \leq Z\), and (2) \(\langle M, Y \cap Z \rangle =\langle M, Y \rangle \cap Z\) for all \(Y \leq G, Z \leq G\) such that \(M \leq Z\). We say that \(H\) is a \textit{generalized modular} subgroup of \(G\) if \(H=AB\) for some modular subgroup \(A\) and subnormal subgroup \(B\) of \(G\). If \(M_n< M_{n-1}< \dots< M_1 < M_{0}=G \), where \(M_i\) is a maximal subgroup of \(M_{i-1}\) for all \(i=1, \dots,n\), then \(M_n \) (\(n > 0\)) is an \(n\)-\textit{maximal subgroup} of \(G\). In this paper, we study finite groups whose \(n\)-maximal subgroups are generalized modular. In particular, we prove the following Theorem 1.3: suppose that \(G\) is soluble and every \(n\)-maximal subgroup of \(G\) is generalized modular. If \(n \leq |\pi (G)|-1\), then \(G\) is supersoluble.Serre weights and Breuil's lattice conjecture in dimension three.https://www.zbmath.org/1452.110662021-02-12T15:23:00+00:00"Le, Daniel"https://www.zbmath.org/authors/?q=ai:le.daniel"Le Hung, Bao V."https://www.zbmath.org/authors/?q=ai:le-hung.bao-v"Levin, Brandon"https://www.zbmath.org/authors/?q=ai:levin.brandon"Morra, Stefano"https://www.zbmath.org/authors/?q=ai:morra.stefanoSummary: We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a \(U(3)\)-arithmetic manifold is purely local, that is, only depends on the Galois representation at places above \(p\). This is a generalization to \(\text{GL}_3\) of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil-Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge-Tate weights \((2,1,0)\) as well as the Serre weight conjectures of \textit{F. Herzig} [Duke Math. J. 149, No. 1, 37--116 (2009; Zbl 1232.11065)] over an unramified field extending the results of
the first author et al. [Invent. Math. 212, No. 1, 1--107 (2018; Zbl 1403.11039)]. We also prove results in modular representation theory about lattices in Deligne-Lusztig representations for the group \(\text{GL}_3(\mathbb{F}_q)\).Affine flag varieties and quantum symmetric pairs.https://www.zbmath.org/1452.170012021-02-12T15:23:00+00:00"Fan, Zhaobing"https://www.zbmath.org/authors/?q=ai:fan.zhaobing"Lai, Chun-Ju"https://www.zbmath.org/authors/?q=ai:lai.chun-ju"Li, Yiqiang"https://www.zbmath.org/authors/?q=ai:li.yiqiang"Luo, Li"https://www.zbmath.org/authors/?q=ai:luo.li"Wang, Weiqiang"https://www.zbmath.org/authors/?q=ai:wang.weiqiangThe quantum groups of finite and affine type \(A\) have a realization in terms of partial flag varieties of finite and affine type \(A\). Recently, it was shown that ``quantum group associated to partial flag varieties of finite type \(B/C\) can be realized as coideal subalgebras of the quantum group of finite type \(A\). The goal of the book is to ``initiate the study of the Schur algebras and quantum groups arising from partial flag varieties of classical affine type beyond type \(A\), generalizing the constructions in finite type \(B/C\). In the book the authors focus focus on the affine type \(C\).
They show ``that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotent) coideal subalgebras of quantum groups of affine \(sl\) and \(gl\) types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotent coideal algebras of affine \(sl\) type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotent quantum affine \(gl\) and its canonical basis.''
The book consists of three parts.
In Part 1 the basic constructions of the affine Schur algebra and its distinguished Lusztig subalgebra (and their ȷȷ, ȷı, ıȷ, ıı-variants) are introduced.
In Part 2 the structures of the family of Lusztig algebras ( and their ȷȷ, ȷı, ıȷ, ıı-variants) is studied. It is shown that they lead to coideal subalgebras \(U^c(\widehat{sl}_n)\) in \(U(\widehat{sl}_n)\).
Part 3 focused on ``the study of the stabilization properties of the family of Schur algebras \(S_{n,d}^c\) ( and their ȷȷ, ȷı, ıȷ, ıı-variants) leading to stabilization algebras which are identified as idempotented coideal subalgebras \(U^c(\widehat{sl}_n)\) of quantum affine \(gl_n\); these stabilization algebras are shown to admit canonical bases (without positivity).''
Reviewer: Dmitry Artamonov (Moskva)On sublattices of the subgroup lattice defined by formation Fitting sets.https://www.zbmath.org/1452.200142021-02-12T15:23:00+00:00"Skiba, Alexander N."https://www.zbmath.org/authors/?q=ai:skiba.alexander-nA set \(\Sigma\) of normal sections \(H/K\) of a finite group \(G\) is called a stratification of \(G\) if the following two properties are satisfied:
\(H/K \in \Sigma\) whenever \(H/K\cong_G T/L\) and \(T/L \in \Sigma;\) \(L/K, H/L \in \Sigma\) for each triple \(K < L < H\) with \(H/K \in \Sigma\)
and \(L \unlhd G.\) Denote with \(L_\Sigma(G)\) the set of all subgroups \(A\) of \(G\) such that \(A^G/A_G \in \Sigma.\) A stratification \(\Sigma\) of \(G\) is called a formation Fitting set of \(G\) provided \(H/(K \cap N) \in \Sigma\) for every two sections \(H/K, H/N \in \Sigma\) and \(HV/K \in \Sigma\) for every two sections \(H/K, V/K \in \Sigma.\) The author proves that \(L_\Sigma(G)\) forms a sublattice of \(L(G)\) for every formation Fitting set \(\Sigma\) of \(G.\) Some application of the lattices of the form \(L_\Sigma(G)\) are given in the theory of finite generalized \(T\)-groups.
Reviewer: Andrea Lucchini (Padova)Decompositions of some Specht modules. I.https://www.zbmath.org/1452.200072021-02-12T15:23:00+00:00"Donkin, Stephen"https://www.zbmath.org/authors/?q=ai:donkin.stephen"Geranios, Haralampos"https://www.zbmath.org/authors/?q=ai:geranios.haralamposSummary: ``We give a decomposition, as a direct sum of indecomposable modules, of several types
of Specht modules in characteristic 2. These include the Specht modules labelled by hooks, whose
decomposability was considered by \textit{G. Murphy} [J. Algebra 66, 156--168 (1980; Zbl 0447.20009)]. Since the main arguments are essentially no more
difficult for Hecke algebras at parameter \(q = - 1\), we proceed in this generality.''
A key tool in the paper is a Schur functor going from representations of a quantum general linear
group to representations of the Hecke algebra.
From the introduction:
``We now suppose \(K\) has characteristic 2. Then, for \(\lambda\) a 2-singular partition, the
Specht module \(\mathrm{Sp}(\lambda)\) may certainly decompose but in general neither a criterion
for decomposability nor the nature of a decomposition as a direct sum of indecomposable components
is known. [\dots]
We here obtain many new families of decomposable Specht modules for Hecke algebras
at parameter \(q =-1\) and describe explicitly their indecomposable components. [\dots]
The key feature which we are able to exploit at \(q =-1\) is that the Schur functor on a tensor
product of symmetric and exterior powers of the natural module is the same as on a
tensor product of the corresponding symmetric powers only.''
Reviewer: Wilberd van der Kallen (Utrecht)Formulas for complexity, invariant measure and RQA characteristics of the period-doubling subshift.https://www.zbmath.org/1452.370132021-02-12T15:23:00+00:00"Poláková, Miroslava"https://www.zbmath.org/authors/?q=ai:polakova.miroslavaSummary: Explicit formulas for complexity and unique invariant measure of the period-doubling subshift can be derived from those for the Thue-Morse subshift, obtained by \textit{S. Brlek} [Discrete Appl. Math. 24, No. 1--3, 83--96 (1989; Zbl 0683.20045)],
\textit{A. de Luca} and \textit{S. Varricchio} [Theor. Comput. Sci. 63, No. 3, 333--348 (1989; Zbl 0671.10050)],
and \textit{F. M. Dekking} [Acta Univ. Carol., Math. Phys. 33, No. 2, 35--40 (1992; Zbl 0790.11017)].
In this note we give direct proofs based on combinatorial properties of the period-doubling sequence. We also derive explicit formulas for correlation integral and two basic characteristics of recurrence quantification analysis (RQA) of the period-doubling subshift: recurrence rate and determinism. As a corollary we obtain that RQA determinism of this subshift converges to 1 as the threshold distance approaches 0.Factorizations of finite groups and related topics.https://www.zbmath.org/1452.200152021-02-12T15:23:00+00:00"Kazarin, Lev"https://www.zbmath.org/authors/?q=ai:kazarin.lev-sIn this survey article, the author explains and integrates classical and recent results related to factorisations of finite groups. Here, we say that a group \(G\) has a (proper) factorisation, or that \(G\) factorises (properly) if and only if there exist (proper) subgroups \(A,B \le G\) such that \(G=A \cdot B\). First, the concept of factorisations is explained and motivated from two perspectives:
1. Under what conditions do factorisations exist, and what do we know about the factors?
2. If \(G\) factorises, then what can we find out about the structure of \(G\) by investigating the structure of the factors?
The author explains non-simplicity criteria using factorisations, distinguishing between results that rely on the CFSG and results that do not.
He also describes the concept of graphs related to groups and how this can be used together with factorisations, for example in order
to characterise certain finite simple groups. This combination of methods also leads to non-simplicity criteria. The next topic is ``triple factorisations'', which means factorisations of the form
\[
G=A \cdot B=B \cdot C=A \cdot C,\text{ where }A,B,C \le G.
\]
After some general results, the author turns to \(ABA\)-groups, which can be seen as generalising factorisations with two factors. If \(A,B \le G\), then \(G\) is an \(ABA\)-group if and only if for all \(g \in G\) there are elements \(a,a_1 \in A\) and \(b \in B\) such that \(g=aba_1\).
Among other things, the author discusses which ones of the sporadic simple groups have factorisations or are \(ABA\)-groups.
The remainder of the article deals with applications of ideas related to factorisations in a more general context, e.g. to nilpotent algebras, nearrings, simply reducible groups and coding theory. Throughout, the author gives a lot of historical context and refers to related work, and the list of references is extensive.
Reviewer: Rebecca Waldecker (Halle)Generic toric ideals and row-factorization matrices in numerical semigroups.https://www.zbmath.org/1452.130302021-02-12T15:23:00+00:00"Eto, Kazufumi"https://www.zbmath.org/authors/?q=ai:eto.kazufumiLet \(S\) be a numerical semigroup minimally generated by \(n_{1},\ldots,n_{s}\) and \(k[S]=k[t^{d}|d \in S]\) be the semigroup ring of \(S\), where \(k\) is a field. The defining ideal of \(S\), denoted by \(I(S)\), is the kernel of the \(k\)-algebra homomorphism \(\phi:k[X_{1},\ldots,X_{s}] \longrightarrow k[S]\) given by \(\phi(x_{i})=t^{n_i}\) for all \(1 \leq i \leq s\). Then \(I(S)\) is a prime binomial ideal, called a toric ideal.
In the paper under review the author provides conditions to check when \(I(S)\) is a generic toric ideal, namely \(I(S)\) has a minimal generating set consisting of binomials with full support. As an application, he proves that if \(k[S]\) is almost Gorenstein and \(s>3\) then \(I(S)\) is not generic.
For the entire collection see [Zbl 1446.20006].
Reviewer: Anargyros Katsabekis (Ankara)Every finite subset of an abelian group is an asymptotic approximate group.https://www.zbmath.org/1452.110152021-02-12T15:23:00+00:00"Nathanson, Melvyn B."https://www.zbmath.org/authors/?q=ai:nathanson.melvyn-bernardLet \(A\) be a non-empty subset (not necessarily finite or symmetric or containing the identity) of a (not necessarily commutative) group \(G\). \(A\) is said to be \((r,\ell)\)-approximate group if there exists a subset \(X\subseteq G\) such that \(|X|\leq \ell\) and \(A^r\subseteq XA\) and \(A\) is asymptotic \((r,\ell)\)-approximate group if \(A^h\) is \((r,\ell)\)-approximate in \(G\) for every sufficiently high \(h\) (\(h,r,\ell\) are positive integers).
In the paper the author studies commutative groups (written additively), however the first result of the paper (Theorem 1) shows that there are finite subsets of groups that are not asymptotic \((r,\ell)\)-approximate group for any integers \(r\geq2\) and \(\ell\geq 1\) (e.g. if \(G\) ia a free group of rank 2 generated by the set \(A=\{a_1,a_2\}\)). On the other hand it is proved (Theorem 6) that every nonempty finite subset of a commutative group is an asymptotic approximate group (this extends the author's result [``Every finite set of integers is an asymptotic approximate group'', Preprint, \url{arXiv:1511.06478}] that every finite set of integers is an asymptotic approximate group). Similarly, it is proved in the paper that every polytope in a real vector space is an asymptotic \((r,\ell)\)-approximate group or that every finite set of lattice points is an asymptotic \((r,\ell)\)-approximate group.
Reviewer: Štefan Porubský (Praha)Compact group actions on topological and noncommutative joins.https://www.zbmath.org/1452.460562021-02-12T15:23:00+00:00"Chirvasitu, Alexandru"https://www.zbmath.org/authors/?q=ai:chirvasitu.alexandru"Passer, Benjamin"https://www.zbmath.org/authors/?q=ai:passer.benjamin-wSummary: We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of \textit{P. F. Baum} et al. [Banach Cent. Publ. 106, 9--18 (2015; Zbl 1343.46064)]: there are no equivariant morphisms \( A \rightarrow A \circledast _\delta H\) or \( H \rightarrow A \circledast _\delta H\), respectively, when \( H\) is a nontrivial compact quantum group acting freely on a unital \(C^\ast\)-algebra \(A\). Here \( A \circledast _\delta H\) denotes the equivariant noncommutative join of \( A\) and \( H\); this join procedure is a modification of the topological join that allows a free action of \( H\) on \( A\) to produce a free action of \( H\) on \( A \circledast _\delta H\). For the classical case \( H = \mathcal {C}(G)\), \( G\) a compact group, we present a reduction of the Type 1 conjecture and counterexamples to the Type 2 conjecture. We also present some examples and conditions under which the Type 2 conjecture does hold.Domination of blocks, fusion systems and hyperfocal subgroups.https://www.zbmath.org/1452.200062021-02-12T15:23:00+00:00"Coconeţ, Tiberiu"https://www.zbmath.org/authors/?q=ai:coconet.tiberiu"Todea, Constantin-Cosmin"https://www.zbmath.org/authors/?q=ai:todea.constantin-cosminLet \(k\) be an algebraically closed field of characteristic \(p>0\), let \(G\) be a finite group, and let \(b\) be a block (idempotent) of the group algebra \(kG\). Moreover, let \((D,e)\) be a maximal \(b\)-Brauer pair, and let \(\mathcal{F}\) denote the corresponding fusion system of \(b\) on the defect group \(D\) of \(b\). The hyperfocal subgroup of \(\mathcal{F}\) is defined by
\[
\mathrm{hyp}(\mathcal{F}) := \langle u \phi(u^{-1}):u \in Q \le D, \; \phi \in \mathrm{O}^p(\mathrm{Aut}_{\mathcal F}(Q))\rangle.
\]
The block \(b\) is called inertial if there exists a basic Morita equivalence between \(b\) and its Brauer correspondent in \(kN_G(D)\). It is called nilpotent if \(\mathrm{hyp}(\mathcal{F}) = 1\).
Now let \(P\) be a normal \(p\)-subgroup of \(G\), and let \(\pi: kG \longrightarrow k{\bar G}\) be the canonical epimorphism where \({\bar G} := G/P\). One of the main results of the paper shows:
\begin{itemize}
\item[(i)] If \(b\) is nilpotent then \(\pi(b)\) is a nilpotent block of \(k{\bar G}\);
\item[(ii)] If \(D = Q \times P\) with \(\mathrm{hyp}(\mathcal{F}) \le Q\) then \(b\) is inertial if and only if \(\pi(b)\) is an inertial block of \(k{\bar G}\).
\end{itemize}
The authors also investigate connections between fusion systems and hyperfocal subgroups of the block \(b\) of \(kG\) and of the block \(\pi(b)\) of \(k{\bar G}\) in the situation where \(G/C_G(P)\) is a \(p\)-group.
Reviewer: Burkhard Külshammer (Jena)Pattern groups and a poset based Hopf monoid.https://www.zbmath.org/1452.200042021-02-12T15:23:00+00:00"Aliniaeifard, Farid"https://www.zbmath.org/authors/?q=ai:aliniaeifard.farid"Thiem, Nathaniel"https://www.zbmath.org/authors/?q=ai:thiem.nathanielA supercharacter theory \(S\) of a finite group G is a pair \((\mathrm{Cl}(S), \mathrm{Ch}(S))\) where \(\mathrm{Cl}(S)\) is a set partition of \(G\) and \(\mathrm{Ch}(S)\) is a set partition of the irreducible characters \(\mathrm{Irr}(G)\) of \(G\), such that:
1) \( \{1\} \in \mathrm{Cl}(S)\)
2) \( |\mathrm{Cl}(S)| = |\mathrm{Ch}(S)|\)
3) For each \(X \in \mathrm{Ch}(S)\) one has \(\chi^X=\sum_{\psi\in X}\psi(1)\psi\in\{\varphi: G\rightarrow \mathbb{C}: \{g,h\}\subset K\in \mathrm{Cl}(S)\text{ implies }\varphi(g)=\varphi(h)\}\)
As the authors write, ``a supercharacter theory of the unipotent upper-triangular matrices of the finite general linear groups gave a representation theoretic interpretation to the Hopf algebra of symmetric functions in non-commuting variables.''
In the present paper, the pattern groups are considered. The patter groups are a family of unipotent groups that are built out of finite posets. The authors use the specific supercharacter theory introduced in [\textit{F. Aliniaeifard}, J. Algebra 469, 464--484 (2017; Zbl 1443.20005)]. Using it, a Hopf monoid naturally arising in the supercharacter theory is investigated. The authors ``give a formula for the coproduct on supercharacters, establish an algebraically independent set of free generators (as a monoid), and construct the primitive elements''.
Reviewer: Dmitry Artamonov (Moskva)The equations of Dirac and Maxwell as a result of combining Minkowski space and the space of orientations into seven-dimensional space-time.https://www.zbmath.org/1452.830242021-02-12T15:23:00+00:00"Sventkovsky, R. A."https://www.zbmath.org/authors/?q=ai:sventkovsky.r-aSummary: The multicomponent wave function of the spin and vector fields is presented as a one-component function depending on the position and orientation of a zero-size moving rotating observer. It is shown that the Dirac and Maxwell equations and the fine structure constant are the result of the connection of two spaces: the Minkowski space and the space of orientations (the observer), and this relationship is not mathematical, but physical in nature.Graphical representations of cyclic permutation groups.https://www.zbmath.org/1452.200012021-02-12T15:23:00+00:00"Grech, Mariusz"https://www.zbmath.org/authors/?q=ai:grech.mariusz"Kisielewicz, Andrzej"https://www.zbmath.org/authors/?q=ai:kisielewicz.andrzej-piotrThe authors obtain a necessary and sufficient condition that a permutation group generated by a single permutation is an automorphism group of an edge-colored graph.
Reviewer: Mohammad-Reza Darafsheh (Tehran)Corrigendum to: ``Effective separability of finitely generated nilpotent groups''.https://www.zbmath.org/1452.200282021-02-12T15:23:00+00:00"Pengitore, Mark"https://www.zbmath.org/authors/?q=ai:pengitore.markSummary: In previous work [New York J. Math. 24, 83--145 (2018; Zbl 1451.20007)], the author claimed a characterization for \(\mathrm{F}_G(n)\) and lower asymptotic bounds for \(\mathrm{Conj}_G(n)\) when \(G\) is a finitely generated nilpotent group. However, a counterexample to the characterization of \(\mathrm{F}_G(n)\) for finitely generated nilpotent groups was communicated to us by Khalid Bou-Rabee which also had consequences to the lower asymptotic bound provided for \(\mathrm{Conj}_G(n)\). The purpose of this note to explain what is incorrect in [loc. cit.] along with the counterexample provided to us. We will also explain what remains correct in [loc. cit.] and how we obtain weaker lower bounds for \(\mathrm{F}_N(n)\) and \(\mathrm{Conj}_N(n)\) which are found in the author's thesis and a forthcoming preprint.Embeddings of maximal tori in classical groups and explicit Brauer-Manin obstruction.https://www.zbmath.org/1452.110372021-02-12T15:23:00+00:00"Bayer-Fluckiger, E."https://www.zbmath.org/authors/?q=ai:bayer-fluckiger.eva"Lee, T.-Y."https://www.zbmath.org/authors/?q=ai:lee.ting-yu"Parimala, R."https://www.zbmath.org/authors/?q=ai:parimala.ramanIn their paper [Comment. Math. Helv. 85, No. 3, 583--645 (2010; Zbl 1223.11047)] \textit{G. Prasad} and \textit{A. S. Rapinchuk} proved a Hasse principle for the existence of an embedding of a global field \(E\) with an involutive automorphism into a simple algebra \(A\) with a given involution \(\tau\) when \(\tau\) is symplectic or when \(\tau\) is orthogonal but \(A \neq M_{2n}(D)\) for a quaternion division algebra.
The first author of this paper had obtained combinatorial criteria for Hasse principle to hold when \(A\) is a matrix algebra and \(\tau\) is orthogonal [J. Eur. Math. Soc. (JEMS) 17, No. 7, 1629--1656 (2015; Zbl 1326.11008)].
Also, building on results of \textit{M. Borovoi} [Math. Ann. 314, No. 3, 491--504 (1999; Zbl 0966.14017)], the second author proved that the Brauer-Manin obstruction is the only obstruction to Hasse principle holding [Comment. Math. Helv. 89, No. 3, 671--717 (2014; Zbl 1321.11043)].
These different points of view are explained by a construction of obstruction to the Hasse principle proved in the paper under review. In particular, the authors define the notion of an `oriented embedding' of a field with involution into a central simple algebra with involution. Using this, they extend the main result of Prasad-Rapinchuk to show that existence of oriented embeddings locally implies a global embedding.
More generally, when \(E\) is an étale algebra with involution \(\sigma\) over a global field \(K\), the authors define a group \(\Sha(E,\sigma)\) closely connected to a Tate-Shafarevich group; this group \(\Sha(E,\sigma)\) encodes ramification properties of the components of \((E, \sigma)\). Associated to oriented embeddings of \((E, \sigma)\) into \((A. \tau)\) locally, the authors define local embedding data which enables them to obtain a homomorphism \(f\) form \(\Sha(E, \sigma)\) to \(\mathbb{Z}/2 \mathbb{Z}\).
The authors' necessary and sufficient criterion for the Hasse principle asserts that given oriented embeddings of \((E, \sigma)\) into \((A, \tau)\) over all completions \(K_v\), there is a global embedding over \(K\) if, and only if, the corresponding \(f : \Sha(E, \sigma) \rightarrow \mathbb{Z}/2 \mathbb{Z}\) is the zero map.
Reviewer: Balasubramanian Sury (Bangalore)Measuring the arcs of the orbit of a one-parameter transformation group.https://www.zbmath.org/1452.510102021-02-12T15:23:00+00:00"Polikanova, Irina Viktorovna"https://www.zbmath.org/authors/?q=ai:polikanova.irina-viktorovnaSummary: In the orbits of the one-parameter transformation group, various geometric structures are introduced by means of bijection with the set of real numbers, in particular, the structure of the Euclidean straight line in the sense used by Hilbert with corresponding concepts: points, half-orbits (an analogue of a ray), an oriented arc (an analogue of a segment with ordered endpoints), the relations of between for three points, the equality of oriented arcs and other invariants of the transformation group. It is shown that the measure of arcs, defined as a positive definite additive function that is invariant with respect to the group of transformations, exists and splits into two independent measures which are uniquely defined on the classes of arcs of a similar orientation by setting the standards -- one in each class -- and coinciding with the measurement results by these standards from different endpoints of the arc. \( \lambda \)-Congruence permits to measure oppositely oriented arcs using a single standard. In this case, the opposite arcs \(ab\) and \(ba\) (not \(\lambda\) congruent) have different measure values. This circumstance forces us to question the correctness of the known proofs of the existence and uniqueness of the measure of the length of a line-segment in Euclidean space. With \(\lambda \)-congruence for \(\lambda=-1\), the orbit becomes a model of Euclidean straight line.Real algebraic links in \(S^3\) and braid group actions on the set of \(n\)-adic integers.https://www.zbmath.org/1452.570042021-02-12T15:23:00+00:00"Bode, Benjamin"https://www.zbmath.org/authors/?q=ai:bode.benjaminThe goal of the paper under review is to construct braid group actions on the set of the \(n\)-adic integers \(\mathbb{Z}_n\). Such study leads to connections between different aspects of the study of braid groups, linking group theoretic properties with the topology of certain configuration spaces and subsets of the space of complex polynomials. Let \(C_n\) be the space of monic complex polynomials of degree \(n\), and \[V_n:=\{(v_1, v_2, \ldots , v_{n-1})\in (\mathbb{C}\backslash\{0\})^{n-1}: v_i\ne v_j \text{ if } i\ne j \}/S_{n-1}.\]
The author proves:
Theorem 1.2. There is a tower of covering spaces
\[\cdots \to Z_{n}^{i+1} \to Z_n^i \to \cdots \to Z_n^2 \to Z_n^1=Z_n\to V_n\simeq D_n\stackrel{p}\to C_n,\]
\noindent where \(p\) is a covering map of degree \(n\), all other arrows are covering maps of degree \(n^{n-1}\) and \(\simeq\) denotes homotopy equivalence.
The fiber over a point \(v\in V_n\) is the set of \(n^{n-1}\)-adic integers \(\mathbb{Z}_{n^{n-1}}\). This provides an action of \(\pi_1(C_n)\), which is the braid group, on the set \(\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}_{n^{n-1}}\), via monodromy. Write this action as \(\phi_n(\ .\ , B):\mathbb{Z}_n \to \mathbb{Z}_n, \ B\in \mathbb{B}_n\).
With very similar considerations, another action \(\psi_n\) of the braid group on \(\mathbb{Z}_{n^n}\cong \mathbb{Z}_n\) is constructed. The author shows:
Theorem 1.3. For both of the constructed actions \(\phi_n\) and \(\psi_n\), the following is true.
\begin{itemize}
\item[(i)] They preserve the metrics on \(\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}_{n^{n-1}}\) and \( \mathbb{Z}_{n^{n}}\), respectively. Therefore, they yield continuous braid group actions on \(\mathbb{Z}_n\).
\item[(ii)] They correspond to sequences of homomorphisms \(\mathcal{B}_n \to S_{n\times (n^{n-1})^j}\) and \(\mathcal{B}_n \to S_{ (n^{n})^j}\), respectively. For \(\phi_n\), the resulting action on \(n\times(n^{n-1})^j\) points is transitive for all \(n\) and \(j\).
\item[(iii)] The kernels \(N_j\) and \(H_j\) of the homomorphisms \(\mathcal{B}_n \to S_{n\times (n^{n-1})^j}\) and \(\mathcal{B}_n \to S_{ (n^{n})^j}\), respectively, form descending series of normal subgroups of the braid group that do not stabilize.
\end{itemize}
The paper contains a quite complete and useful exposition of the background which is used in the paper concerning the \(n\)-adic integers, polynomials, braids and covering maps.
It is explained how this material can be used to improve invariants of braids or conjugacy classes. Also the sequences of normal subgroups that are given by the kernels of the homomorphisms given in the statement of Theorem 1.3 (ii) are studied, and it is shown that they do not stabilize. Calculation with the action \(\psi_n\) is used to show an infinite family of braids close to real algebraic links. The author obtains the following result:
Theorem 1.4. Let \(\epsilon \in \{\pm 1\}\) and let \(B=\Pi_{j+1}^{\ell}\omega_{i_j}^{\epsilon}\) with \(i_j \in \{1,2,\ldots,5\}\) be a \(3\)-strand braid with \[\omega_1=\sigma_2, \ \ \ \ \omega_2=\sigma_1^2, \ \ \ \ \omega_3=(\sigma_1\sigma_2\sigma_1)^2,\] \[\omega_4=(\sigma_2\sigma_1\sigma_2^{-1}\sigma_1\sigma_2)^2, \ \ \ \ \omega_5=\sigma_2^{-1}\sigma_1\sigma_2\sigma_2\sigma_1,\] and such that there is a \(j\) such that \(i_j=3\) or such that there is only one residue class \(k\) mod \(3\) such that \(i_j\ne k\) for all \(j=1,2,\ldots, \ell\). Then, the closure of \(B^2\) is real algebraic.
Reviewer: Daciberg Lima Gonçalves (São Paulo)On the identification of finite non-group semigroups of a given order.https://www.zbmath.org/1452.200542021-02-12T15:23:00+00:00"Monsef, M."https://www.zbmath.org/authors/?q=ai:monsef.m-e-abdel"Doostie, H."https://www.zbmath.org/authors/?q=ai:doostie.hosseinSummary: Identifying finite non-group semigroups for every positive integer is significant because of many applications of such semigroups are functional in various branches of sciences such as computer science, mathematics and finite machines. The finite non-commutative monoids as a type of such semigroups were identified in 2014, for every positive integer. We here attempt to identify the finite \textit{commutative monoids} and finite \textit{commutative non-monoids} of a given integer \(n=p^\alpha q^\beta \), for every integers \(\alpha , \beta \ge 2\) and different primes \(p\) and \(q\). In order to recognize the commutative monoids, we present a class of 2-generated monoids of a given order, and for the commutative non-monoids of order \(n=p^\alpha q^\beta,\) we give the minimal generating set. Moreover, we prove that there are exactly \((p^\alpha -2)(q^\beta -2)\) non-isomorphic commutative non-monoids of order \(p^\alpha q^\beta\). The identification of non-group semigroups for the integers \(p^{2\alpha}\) and \(2p^\alpha\) is achieved. The automorphism groups of these groups are specified as well. As a result of this study, an interesting difference between the abelian groups and the commutative semigroups of order \(p^2\) is presented.Nielsen realization by gluing: limit groups and free products.https://www.zbmath.org/1452.200202021-02-12T15:23:00+00:00"Hensel, Sebastian"https://www.zbmath.org/authors/?q=ai:hensel.sebastian-wolfgang"Kielak, Dawid"https://www.zbmath.org/authors/?q=ai:kielak.dawidSummary: We generalize the Karrass-Pietrowski-Solitar and the Nielsen realization theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of \textit{M. Handel} and \textit{L. Mosher} [Geom. Topol. 17, No. 3, 1581--1672 (2013; Zbl 1278.20053)] and on the outer space of a free product of \textit{V. Guirardel} and \textit{G. Levitt} [Proc. Lond. Math. Soc. (3) 94, No. 3, 695--714 (2007; Zbl 1168.20011)], and also a relative version of the Nielsen realization theorem, which, in the case of free groups, answers a question of Karen Vogtmann. We also prove Nielsen realization for limit groups and, as a byproduct, obtain a new proof that limit groups are CAT(\(0\)).
The proofs rely on a new version of Stallings' theorem on groups with at least two ends, in which some control over the behavior of virtual free factors is gained.On the monoid of cofinite partial isometries of \(\mathbb{N}^n\) with the usual metric.https://www.zbmath.org/1452.200582021-02-12T15:23:00+00:00"Gutik, Oleg"https://www.zbmath.org/authors/?q=ai:gutik.oleg-v"Savchuk, Anatolii"https://www.zbmath.org/authors/?q=ai:savchuk.anatoliiSummary: In this paper we study the structure of the monoid \(\mathbf{I}\mathbb{N}^n_\infty\) of cofinite partial isometries of the \(n\)-th power of the set of positive integers \(\mathbb{N}\) with the usual metric for a positive integer \(n\geq2\). We describe the group of units and the subset of idempotents of the semigroup \(\mathbf{I}\mathbb{N}^n_\infty\), the natural partial order and Green's relations on \(\mathbf{I}\mathbb{N}^n_\infty\). In particular we show that the quotient semigroup \(\mathbf{I}\mathbb{N}^n_\infty/\mathfrak{C}_{\mathrm{mg}}\), where \(\mathfrak{C}_{\mathrm{mg}}\) is the minimum group congruence on \(\mathbf{I}\mathbb{N}^n_\infty\), is isomorphic to the symmetric group \(\mathscr{S}_n\) and \(\mathscr{D}=\mathscr{J}\) in \(\mathbf{I}\mathbb{N}^n_\infty\). Also, we prove that for any integer \(n\geq2\) the semigroup \(\mathbf{I}\mathbb{N}^n_\infty\) is isomorphic to the semidirect product \(\mathscr{S}_n \ltimes (\mathscr{P}_\infty(\mathscr{N}^n),\cup)\) of the free semilattice with the unit \((\mathscr{P}_\infty(\mathbb{N}^n),\cup)\) by the symmetric group \(\mathscr{S}_n\).Geodesic Gaussian integer continued fractions.https://www.zbmath.org/1452.110842021-02-12T15:23:00+00:00"Hockman, Meira"https://www.zbmath.org/authors/?q=ai:hockman.meiraThis article deals with geodesic continued fractions with Gaussian integer coefficients. The author provides the Farey graph which is modeled by the realization in \(H^3\) where Farey neighbors are joined by hyperbolic geodesics as seen in the Farey tessellation of \(H^3\) by Farey octahedrons. The author introduces the natural distance on the set of the vertices of the Farey graph and study the structure of the Farey neighbourhoods of reduced Gaussian rational numbers (e.g., the vertices of the Farey graph). The conditions for a path to be a geodesic path are shown. Finally the conditions for the existence of an infinite geodesic Gaussian integer continued fraction and further extension to integer quaternion entries are discussed.
Reviewer: Oleg Karpenkov (Liverpool)Palindromic automorphisms of right-angled Artin groups.https://www.zbmath.org/1452.200292021-02-12T15:23:00+00:00"Fullarton, Neil J."https://www.zbmath.org/authors/?q=ai:fullarton.neil-j"Thomas, Anne"https://www.zbmath.org/authors/?q=ai:thomas.anne|thomas.anne.1Summary: We introduce the palindromic automorphism group and the palindromic Torelli group of a right-angled Artin group \(A_\Gamma\). The palindromic automorphism group \(\Pi\mathrm{A}_\Gamma\) is related to the principal congruence subgroups of \(\mathrm{GL}(n,\mathbb{Z})\) and to the hyperelliptic mapping class group of an oriented surface, and sits inside the centraliser of a certain hyperelliptic involution in \(\mathrm{Aut}(A_\Gamma)\). We obtain finite generating sets for \(\Pi\mathrm{A}_\Gamma\) and for this centraliser, and determine precisely when these two groups coincide. We also find generators for the palindromic Torelli group.Approximate lattices.https://www.zbmath.org/1452.220022021-02-12T15:23:00+00:00"Björklund, Michael"https://www.zbmath.org/authors/?q=ai:bjorklund.michael"Hartnick, Tobias"https://www.zbmath.org/authors/?q=ai:hartnick.tobiasSummary: In this article we introduce and study uniform and nonuniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense of Tao) which simultaneously generalize lattices in lcsc group and mathematical quasicrystals (Meyer sets) in lcsc abelian groups. We show that envelopes of strong approximate lattices are unimodular and that approximate lattices in nilpotent groups are uniform. We also establish several results relating properties of approximate lattices and their envelopes. For example, we prove a version of the Milnor-Schwarz lemma for uniform approximate lattices in compactly generated lcsc groups, which we then use to relate the metric amenability of uniform approximate lattices to the amenability of the envelope. Finally we extend a theorem of \textit{B. Kleiner} and \textit{B. Leeb} [Publ. Math., Inst. Hautes Étud. Sci. 86, 115--197 (1997; Zbl 0910.53035)] to show that the isometry groups of irreducible higher-rank symmetric spaces of noncompact type are quasi-isometrically rigid with respect to finitely generated approximate groups.On 2-Verma modules for quantum \({\mathfrak {sl}_{2}}\).https://www.zbmath.org/1452.170202021-02-12T15:23:00+00:00"Naisse, Grégoire"https://www.zbmath.org/authors/?q=ai:naisse.gregoire"Vaz, Pedro"https://www.zbmath.org/authors/?q=ai:vaz.pedroSummary: In this paper we study the superalgebra \(A_n\), introduced by the authors in previous work on categorification of Verma modules for quantum \({\mathfrak {sl}_{2}}\) [Proc. Lond. Math. Soc. (3) 117, No. 6, 1181--1241 (2018; Zbl 07000621)]. The superalgebra \(A_n\) is akin to the nilHecke algebra, and shares similar properties. We also prove a uniqueness result about 2-Verma modules on \(\Bbbk \)-linear 2-categories.Generating sequences and semigroups of valuations on 2-dimensional normal local rings.https://www.zbmath.org/1452.130052021-02-12T15:23:00+00:00"Dutta, Arpan"https://www.zbmath.org/authors/?q=ai:dutta.arpanLet \(\alpha\) and \(\beta\) be primitive \(m\)-th root and \(n\)-th root of unity, respectively, let \(K\) be an algebraically close field of characteristic zero and let \(R=K[X,Y]\) be a polynomial ring in two variables over \(K\) with \(m=(X,Y)\) its maximal ideal. The author gives a characterization for the existence of a rational rank one non discrete valuation dominating \(R_m\) with a generating sequence of eigenfunctions for a subgroup, \(H\), of \(\langle\alpha\rangle\times\langle\beta\rangle\) using the greatest common divisor of \(m\) and \(n\).
He also gives a method for computing the semigroup of values of elements of \(K[X,Y]^H\). Finally, he shows that this kind of valuation restricted to the quotient field of a local domain does not split in \(K[X,Y]_m\).
The paper es quite technical so a good backgroud is required to fully understand it.
Reviewer: Daniel Marín Aragon (Cádiz)Representations of the necklace braid group: topological and combinatorial approaches.https://www.zbmath.org/1452.200322021-02-12T15:23:00+00:00"Bullivant, Alex"https://www.zbmath.org/authors/?q=ai:bullivant.alex"Kimball, Andrew"https://www.zbmath.org/authors/?q=ai:kimball.andrew"Martin, Paul"https://www.zbmath.org/authors/?q=ai:martin.paul-purdon"Rowell, Eric C."https://www.zbmath.org/authors/?q=ai:rowell.eric-cThe necklace braid group (of rank \(n\)) is the motion group of the \(n\)-necklace which consists of \(n\) pairwise unlinked circles each linked to an auxiliary circle. Alternatively, the necklace group can be seen as an extension of the braid group by one extra generator (in the sense that it contains the braid group, and the quotient is generated by one element), namely, that corresponds to rotating the auxilary circle in the necklace. By the latter description, given a braid group representation, one can try to extend it to a representation of the necklace braid group by finding an appropriate action of the extra generator.
Constructed in the paper are so-called standard extensions, which exist for all irreducible braid group representations. The paper then discusses standard and non-standard extensions of various braid group representaions: the standard representation; the Burau representation; the Lawrence-Krammer-Bigelow representation; and some local representations, e.g., the representations coming from a braided fusion category, as well as a physics aspect of certain standard extensions.
The paper also describes a relation (a group morphism) between the necklace braid group and the loop braid group and compares some of the above results with the case for the loop braid group.
Reviewer: Hankyung Ko (Uppsala)A new type of fuzzy quasi-ideals of ordered semigroups.https://www.zbmath.org/1452.060122021-02-12T15:23:00+00:00"Cristea, Irina"https://www.zbmath.org/authors/?q=ai:cristea.irina"Mahboob, Ahsan"https://www.zbmath.org/authors/?q=ai:mahboob.ahsan"Khan, Noor Mohammad"https://www.zbmath.org/authors/?q=ai:khan.noor-mohammadSummary: In this paper, we generalize the concept of an \((\epsilon,\epsilon \vee q_k)\)-fuzzy quasi-ideal of an ordered semigroup, using the notion of \((k^\ast, q)\)-quasi-coincidence of an ordered fuzzy point with a fuzzy subset of the support ordered semigroup. First we define and characterize in different ways the \((\epsilon,\epsilon\vee (k^\ast,q_k))\)-fuzzy quasi-ideals of an ordered semigroup. Secondly, we present some relationships between these generalized fuzzy quasi-ideals and similar generalizations of fuzzy left/right ideals or fuzzy bi-ideals (based on the \((k^\ast,q)\)-quasi-coincidence relation). Similarities with quasi-ideals in ordered semigroups are discussed at the end of the paper.Character sheaves on neutrally solvable groups.https://www.zbmath.org/1452.200422021-02-12T15:23:00+00:00"Deshpande, Tanmay"https://www.zbmath.org/authors/?q=ai:deshpande.tanmaySummary: Let \( G\) be an algebraic group over an algebraically closed field \( \mathsf {k}\) of characteristic \( p>0\). In this paper we develop the theory of character sheaves on groups \( G\) such that their neutral connected components \( G^\circ \) are solvable algebraic groups. For such algebraic groups \( G\) (which we call neutrally solvable) we will define the set \( \operatorname {CS}(G)\) of character sheaves on \( G\) as certain special (isomorphism classes of) objects in the category \( \mathscr {D}_G(G)\) of \( G\)-equivariant \( \overline {\mathbb{Q}}_{\ell }\)-complexes (where we fix a prime \( \ell \neq p\)) on \( G\). We will describe a partition of the set \( \operatorname {CS}(G)\) into finite sets known as \( \mathbb{L}\)-packets and we will associate a modular category \( \mathscr {M}_L\) with each \( \mathbb{L}\)-packet \( L\) of character sheaves using a truncated version of convolution of character sheaves. In the case where \( \mathsf {k}=\overline {\mathbb{F}}_q\) and \( G\) is equipped with an \( \mathbb{F}_q\)-Frobenius \( F\) we will study the relationship between \( F\)-stable character sheaves on \( G\) and the irreducible characters of (all pure inner forms of) \( G^F\). In particular, we will prove that the notion of almost characters (introduced by \textit{T. Shoji} [J. Algebra 145, No. 2, 468--524 (1992; Zbl 0744.20039)] using Shintani descent) is well defined for neutrally solvable groups and that these almost characters coincide with the ``trace of Frobenius'' functions associated with \( F\)-stable character sheaves. We will also prove that the matrix relating the irreducible characters and almost characters is block diagonal where the blocks on the diagonal are parametrized by \( F\)-stable \( \mathbb{L}\)-packets. Moreover, we will prove that the block in this transition matrix corresponding to any \( F\)-stable \( \mathbb{L}\)-packet \( L\) can be described as the crossed S-matrix associated with the auto-equivalence of the modular category \( \mathscr {M}_L\) induced by \( F\).Generalization of Knuth's formula for the number of skew tableaux.https://www.zbmath.org/1452.051902021-02-12T15:23:00+00:00"Na, Minwon"https://www.zbmath.org/authors/?q=ai:na.minwonSummary: We take an elementary approach to derive a generalization of Knuth's formula using Lassalle's explicit formula. In particular, we give a formula for the Kostka numbers of a shape \(\mu \vdash n\) and weight \((m, 1^{n-m})\) for \(m = 3, 4\).Exponent of a finite group admitting a coprime automorphism.https://www.zbmath.org/1452.200172021-02-12T15:23:00+00:00"Rodrigues, Sara"https://www.zbmath.org/authors/?q=ai:rodrigues.sara"Shumyatsky, Pavel"https://www.zbmath.org/authors/?q=ai:shumyatsky.pavelLet \(G\) be a finite group, let \(\phi \in\mathrm{Aut}(G)\) and let \(F_{\phi}=C_G(\phi)\) and \(K_{\phi}=\{x^{-1}x^{\phi} \mid x\in G\}.\) The authors are interested to bound the exponent of \(G\) and of \([G, \phi]=\langle K_{\phi} \rangle\) given that some local information is at hand.
They obtain two main theorems as follows:
Theorem 1.2: Let \(n=|\phi|\) be co-prime to \(|G|\) and suppose that every element in the set \(F_{\phi}\cup K_{\phi}\) is contained in a \(\phi\)-invariant subgroup of \(G\) of exponent dividing \(e\). Then, then exponent of \(G\) is bounded in terms of \(e\) and \(n\).
Theorem 1.3: Let \(n=|\phi|\) be co-prime to \(|G|\), with \(F_{\phi}\) nilpotent of class \(c\) and with \(x^e=1\) for all \(x\in K_{\phi}.\) Suppose that any two elements of \(K_{\phi}\) are contained in a \(\phi\)-invariant solvable subgroup of \(G\) of derived length \(d\). Then, the exponent of \([G, \phi]\) is bounded in terms of \(c,d,e,n\).
The proof of Theorem 1.2 uses Lie-theoretical tools, while the proof of Theorem 1.3 depends on the classification of the finite simple groups (which forces \(G\) to be solvable).
A result of independent interest is Lemma 2.3, which gives necessary and sufficient conditions for having \(G=K_{\phi}F_{\phi}\) for a co-prime automorphism \(\phi\) of \(G\).
Reviewer: Marian Deaconescu (Safat)Transition formulas for involution Schubert polynomials.https://www.zbmath.org/1452.200022021-02-12T15:23:00+00:00"Hamaker, Zachary"https://www.zbmath.org/authors/?q=ai:hamaker.zachary"Marberg, Eric"https://www.zbmath.org/authors/?q=ai:marberg.eric"Pawlowski, Brendan"https://www.zbmath.org/authors/?q=ai:pawlowski.brendanSummary: The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group \(S_n\). Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote as \({\hat{\mathfrak S}}_y\) (to be called \textit{involution Schubert polynomials}) and \(\hat{\mathfrak S}^{\mathtt{FPF}}_y\) (to be called \textit{fixed-point-free involution Schubert polynomials}). Our main results are explicit formulas decomposing the product of \({\hat{\mathfrak S}}_y\) (respectively, \(\hat{\mathfrak S}^{\mathtt{FPF}}_y\)) with any \(y\)-invariant linear polynomial as a linear combination of other involution Schubert polynomials. These identities serve as analogues of Lascoux and Schützenberger's transition formula for Schubert polynomials, and lead to a self-contained algebraic proof of the nontrivial equivalence of several definitions of \({\hat{\mathfrak S}}_y\) and \( \hat{\mathfrak S}^{\mathtt{FPF}}_y\) appearing in the literature. Our formulas also imply combinatorial identities about \textit{involution words}, certain variations of reduced words for involutions in \(S_n\). We construct operators on involution words based on the Little map to prove these identities bijectively. The proofs of our main theorems depend on some new technical results, extending work of Incitti, about covering relations in the Bruhat order of \(S_n\) restricted to involutions.On generalization of the notion of Moufang loop to \(n\)-ary case.https://www.zbmath.org/1452.200632021-02-12T15:23:00+00:00"Onoi, V. I."https://www.zbmath.org/authors/?q=ai:onoi.v-i.1"Ursu, L. A."https://www.zbmath.org/authors/?q=ai:ursu.leonid-aSummary: Using isotopical approach we generalize concept of binary Moufang loop on \(n\)-ary \((n>2)\) case. We give two examples of ternary Moufang loop which are not a ternary group.Representation stability for filtrations of Torelli groups.https://www.zbmath.org/1452.180042021-02-12T15:23:00+00:00"Patzt, Peter"https://www.zbmath.org/authors/?q=ai:patzt.peterDieser Artikel studiert die Struktur gewisser Funktoren von der Kategorie \(VIC_\mathbb{Q}\) oder \(SI_\mathbb{Q}\) nach den Vektorräumen über \(\mathbb{Q}\). Hier bezeichnet \(VIC_\mathbb{Q}\) (beziehungsweise \(SI_\mathbb{Q}\)) die Kategorie, deren Objekte die endlich-dimensionalen Vektorräumen (bzw. die endlich-dimensionalen symplektischen Räumen) über \(\mathbb{Q}\) sind, und deren Pfeilen die linearen Monomorphismen mit einer gegebenen Retraktion (bzw. die symplektischen Abbildungen) sind. Solch ein Funktor wird kurz einen \(VIC_\mathbb{Q}\)-Modul oder \(SI_\mathbb{Q}\)-Modul genannt.
Die Bewertung eines \(\text{VIC}_\mathbb{Q}\)-Moduls (bzw. \(SI_\mathbb{Q}\)-Moduls) \(F\) über \(\mathbb{Q}^n\) (bzw. \(\mathbb{Q}^{2n}\)) ist natürlich eine Darstellung über \(\mathbb{Q}\) der linearen Gruppe \(\mathrm{GL}_n(\mathbb{Q})\) (bzw. der symplektischen Gruppe \(\mathrm{Sp}_{2n}(\mathbb{Q})\)). Falls \(F\) einige Endlichkeitseigenschafen (zum Beispiel: \(F\) endlich erzeugt) besitzt, ist die Darstellungensequenz \((F(\mathbb{Q}^n))\) (bzw. \((F(\mathbb{Q}^{2n}))\)) gar nicht beliebig. Die Studie von Phänomenen dieser Art wird \textit{Darstellungsstabilität} (\textit{Representation stability} auf Englisch) genannt.
Die Kategorie \textit{aller} Darstellungen (sogar in der Charakteristik \(0\)) der Gruppe \(\mathrm{GL}_n(\mathbb{Q})\) oder \(\mathrm{Sp}_{2n}(\mathbb{Q})\) ist schrecklich für \(n\ge 2\) (diese Gruppen enthalten eine nicht-kommutative freie Gruppe). Der Autor betrachtet hier nur Funktoren deren Werten \textit{rationale} Darstellungen der algebraischen Gruppen \(\mathrm{GL}_n(\mathbb{Q})\) oder \(\mathrm{Sp}_{2n}(\mathbb{Q})\) sind. Diese Funktoren werden \textit{rational} genannt. Diese Bedingung ist vernünftig, insofern als alle Darstellungen von \(SL_n(\mathbb{Q})\) (das gilt \textit{nicht} für \(\mathrm{GL}_n(\mathbb{Q})\)), die endlich-dimensional über \(\mathbb{Q}\) sind, rational sind (Siehe [\textit{R. Steinberg}, Contemp. Math. 45, 335--350 (1985; Zbl 0579.20038)]). Noch dazu sind die rationalen Darstellungen von \(\mathrm{GL}_n(\mathbb{Q})\) oder \(\mathrm{Sp}_{2n}(\mathbb{Q})\) halb-einfach und wohl verstanden.
Die theoretischen Hauptergebnisse des Artikels zeigen, daß endlich-erzeugte rationale \(\text{VIC}_\mathbb{Q}\)- oder \(SI_\mathbb{Q}\)-Moduln noethersch sind, und daß ihre Werte ein regelmässiges Verhalten als Darstellungen von \(\mathrm{GL}_n(\mathbb{Q})\) oder \(\mathrm{Sp}_{2n}(\mathbb{Q})\) haben (was genau mit Partitionen sich ausdrücken läßt). Das wird dann auf die Rationalisierung merkwürdiger Filtrierungen der Gruppen \(IA_n:=\mathrm{Ker}\,\big(\mathrm{Aut}(F_n)\to \mathrm{GL}_n(\mathbb{Z})\big)\) (von der Abelianisierung einer freien Gruppe induziert) und Torellis Gruppen angewandt. Diese Filtrierungen liefern \(\text{VIC}_\mathbb{Z}\)-Moduln bzw \(SI_\mathbb{Z}\)-Moduln (mit dem Grundring \(\mathbb{Z}\) statt \(\mathbb{Q}\) zur Quelle), aber der Autor zeigt, dass diese Moduln natürlich als rationale \(\text{VIC}_\mathbb{Q}\)- oder \(SI_\mathbb{Q}\)-Moduln sich erweitern lassen.
Erwähnen wir einige verwandte Ergebnisse: der Autor dieses Referats [Fundam. Math. 233, No. 3, 197--256 (2016; Zbl 1353.18001)] bewies (mit unabhängigen Methoden, die den Begriff von \textit{polynomischen Funktoren} benutzen) qualitative Eigenschaften der Quotiente der selben Filtrierungen der Gruppen \(IA_n\), die auch über die ganzen Zahlen gelten, aber die weniger genaue Auskünfte über \(\mathbb{Q}\) geben. In einer schönen Vorveröffentlichung [``Effective and infinite-rank superrigidity in the context of representation stability'', Preprint, \url{arXiv:1902.05603}] beweist \textit{N. Harman} interessante Struktureigenschaften der endlich-erzeugten Funktoren von \(\text{VIC}_\mathbb{Z}\) nach \textit{endlich-dimensionalen} Vektorräumen über \(\mathbb{C}\).
Reviewer: Aurelien Djament (Villeneuve d'Ascq)On the center of the group of quasi-isometries of the real line.https://www.zbmath.org/1452.200362021-02-12T15:23:00+00:00"Chakraborty, Prateep"https://www.zbmath.org/authors/?q=ai:chakraborty.prateepSummary: Let \(QI ( \mathbb{R} )\) denote the group of all quasi-isometries \(f : \mathbb{R} \rightarrow \mathbb{R} \). Let \(Q_+\)(and \(Q_-)\) denote the subgroup of \(QI ( \mathbb{R} )\) consisting of elements which are identity near \(- \infty \) (resp. \(+ \infty )\). We denote by \(QI^+(\mathbb{R} )\) the index 2 subgroup of \(QI ( \mathbb{R} )\) that fixes the ends \(+ \infty, - \infty \). We show that \(QI^+(\mathbb{R}) \cong Q_+ \times Q_-\). Using this we show that the center of the group \(QI ( \mathbb{R} )\) is trivial.The degree of commutativity and lamplighter groups.https://www.zbmath.org/1452.200662021-02-12T15:23:00+00:00"Cox, Charles Garnet"https://www.zbmath.org/authors/?q=ai:cox.charles-garnetAlgebraically compact abelian \(TI\)-groups.https://www.zbmath.org/1452.200532021-02-12T15:23:00+00:00"Kompantseva, Ekaterina Igorevna"https://www.zbmath.org/authors/?q=ai:kompantseva.ekaterina-igorevna"Nguyen, T. Q. T."https://www.zbmath.org/authors/?q=ai:nguyen.t-q-tSummary: An abelian group \(G\) is called a \(TI\)-group if every associative ring with additive group \(G\) is filial. An abelian group \(G\) such that every (associative) ring with additive group \(G\) is an \(SI\)-ring (a hamiltonian ring) is called an \(SI\)-group (an \(SI_H\)-group). In this paper, \(TI\)-groups, as well as \(SI\)-groups and \(SI_H\)-groups are described in the class of reduced algebraically compact abelian groups.The Wells exact sequence for the automorphism group of a Lie ring extension.https://www.zbmath.org/1452.170222021-02-12T15:23:00+00:00"Jamali, A. R."https://www.zbmath.org/authors/?q=ai:jamali.a-r-m-j-uThe normal complement problem and the structure of the unitary subgroup.https://www.zbmath.org/1452.160352021-02-12T15:23:00+00:00"Kaur, Surinder"https://www.zbmath.org/authors/?q=ai:kaur.surinder"Khan, Manju"https://www.zbmath.org/authors/?q=ai:khan.manjuThe study of the structure of the unit group and its unitary subgroup is one of the hardest and most interesting problems in modular group algebras. In 1977, \textit{R. K. Dennis} posed the question for which group \(G\) and ring \(R\), \(G\) has a normal complement in the unit group \(U(RG)\) of the group ring \(RG\) (see reference [in: Ring Theory II, Proc. 2nd Okla. Conf. 1975, 103--130 (1977; Zbl 0357.16003)]). For modular group algebras of finite \(p\)-groups, the normal complement problem has been resolved affirmatively for finite abelian \(p\)-groups, finite \(p\)-groups of exponent \(p\) with nilpotency class \(2\) and for some finite \(2\)-groups (cf. [\textit{L. E. Moran} and \textit{R. N. Tench}, Isr. J. Math. 27, 331--338 (1977; Zbl 0374.20005)]). For a more detailed study of this problem, one can refer to [\textit{A. Bovdi}, Publ. Math. 52, No. 1--2, 193--244 (1998; Zbl 0906.16016); \textit{L. R. Ivory}, Proc. Am. Math. Soc. 79, 9--12 (1980; Zbl 0401.20018); \textit{D. L. Johnson}, Proc. Am. Math. Soc. 68, 19--22 (1978; Zbl 0264.20020)]. Some results on this problem for modular group algebras of finite groups which are not \(p\)-groups can be found in the bibliography cited at the end of the article, and especially in [\textit{K. Kaur} et al., J. Algebra Appl. 16, No. 1, Article ID 1750011, 11 p. (2017; Zbl 1358.16021)] and [\textit{S. Kaur} and \textit{M. Khan}, Commun. Algebra 47, No. 9, 3842--3848 (2019; Zbl 07076207)], respectively.
The authors of the present work prove three theorems, namely Theorems 1, 2 and 3. The last two ones have more technical character, while Theorem 1, which states that if \(p\) is an odd prime, \(F\) is the field with \(p\) elements and \(G\) is a finite split metabelian
\(p\)-group of exponent \(p\), then \(G\) has a normal complement in the normalized unit group \(V(FG)\), is rather more conceptual.
Reviewer: Peter Danchev (Sofia)Intersections of subgroups in virtually free groups and virtually free products.https://www.zbmath.org/1452.200212021-02-12T15:23:00+00:00"Klyachko, Anton A."https://www.zbmath.org/authors/?q=ai:klyachko.anton-a"Ponfilenko, Anastasia N."https://www.zbmath.org/authors/?q=ai:ponfilenko.anastasia-nThe authors prove the following generalisation of the Friedman-Mineyev theorem (earlier known as the Hanna Neumann conjecture): if \(A\) and \(B\) are nontrivial free subgroups of any group containing a free subgroup of index \(n\), then \(\mbox{rank}(A \cap B) - 1 \leq n (\mbox{rank}(A) - 1) (\mbox{rank}(B) - 1)\). Also they show that for any natural numbers \(k, l, n\) there exists a group \(G\) containing free subgroups \(A\), \(B\) and \(F\) such that rank\((A) = k\), rank\((B) = l\), \(|G:F| = n\) and this inequality is an equality.
Reviewer: Alexander Ivanovich Budkin (Barnaul)Combinatorics of extended affine root systems (type \(A_1\)).https://www.zbmath.org/1452.170142021-02-12T15:23:00+00:00"Azam, Saeid"https://www.zbmath.org/authors/?q=ai:azam.saeid"Kharaghani, Zahra"https://www.zbmath.org/authors/?q=ai:kharaghani.zahra\(\text{PGL}(2,\mathbb{F}_q)\) acting on \(\mathbb{F}_q(x)\).https://www.zbmath.org/1452.110382021-02-12T15:23:00+00:00"Hou, Xiang-Dong"https://www.zbmath.org/authors/?q=ai:hou.xiang-dongSummary: Let \(\mathbb{F}_q(x)\) be the field of rational functions over \(\mathbb{F}_q\) and treat \(\text{PGL}(2, \mathbb{F}_q)\) as the group of degree one rational functions in \(\mathbb{F}_q(x)\) equipped with composition. \(\text{PGL}(2,\mathbb{F}_q)\) acts on \(\mathbb{F}_q(x)\) from the right through composition. The Galois correspondence and Lüroth's theorem imply that every subgroup \(H\) of \(\text{PGL}(2,\mathbb{F}_q)\) is the stabilizer of some rational function \(\pi_H(x) \in \mathbb{F}_q(x)\) with \(\mathrm{deg} \pi_H = |H|\) under this action, where \(\pi_H(x)\) is uniquely determined by \(H\) up to a left composition by an element of \(\text{PGL}(2,\mathbb{F}_q)\). In this article, we determine the rational function \(\pi_H(x)\) explicitly for every \(H < \text{ PGL}(2,\mathbb{F}_q)\).Growth of periodic Grigorchuk groups.https://www.zbmath.org/1452.200402021-02-12T15:23:00+00:00"Erschler, Anna"https://www.zbmath.org/authors/?q=ai:erschler.anna"Zheng, Tianyi"https://www.zbmath.org/authors/?q=ai:zheng.tianyiThe authors construct random walks on torsion Grigorchuk groups, with finite entropy and power-law tail decay, having non-trivial Poisson boundary. It is known that, for a finitely generated group, there exists a quantitative relation between its growth function and the tail decay of a measure corresponding to such a random walk [\textit{V. A. Kaimanovich} and \textit{A. M. Vershik}, Ann. Probab. 11, 457--490 (1983; Zbl 0641.60009); \textit{Y. Derriennic}, Asterisque 74, 183--201 (1980; Zbl 0446.60059)]. Using the above random walks, the authors obtain close to the optimal lower volume estimates for these groups. In particular, for the first Grigorchuk group and certain more general cases, they found explicit asymptotic formulas for the volume function.
Reviewer: Anatoly N. Kochubei (Kyïv)Verbally closed virtually free subgroups.https://www.zbmath.org/1452.200412021-02-12T15:23:00+00:00"Klyachko, Ant. A."https://www.zbmath.org/authors/?q=ai:klyachko.anton-a"Mazhuga, A. M."https://www.zbmath.org/authors/?q=ai:mazhuga.andrey-mFully inert subgroups of torsion-complete \(p\)-groups.https://www.zbmath.org/1452.200522021-02-12T15:23:00+00:00"Goldsmith, Brendan"https://www.zbmath.org/authors/?q=ai:goldsmith.brendan"Salce, Luigi"https://www.zbmath.org/authors/?q=ai:salce.luigiA subgroup \(H\) of an abelian group \(G\) is called fully inert if the quotient \((H+\phi(H))/H\) is finite for every endomorphism \(\phi\) of \(G\). This is a common generalization of the notions of fully invariant, finite and finite-index subgroups. Two subgroups \(K\) and \(L\) of \(G\) are commensurable if \([K:L\cap K]\) and \([L:L\cap K]\) are both finite. \(\phi\)-inert subgroups were a basic tool in the definition of intrinsic algebraic entropy introduced in [\textit{D. Dikranjan} et al., J. Pure Appl. Algebra 219, No. 7, 2933--2961 (2015; Zbl 1155.20041)], which in turn is a variant of the notion of algebraic entropy that was investigated in depth in [\textit{D. Dikranjan} et al., Trans. Am. Math. Soc. 361, No. 7, 3401--3434 (2009; Zbl 1176.20057)]. The authors states that fully inert subgroups of torsion-complete abelian \(p\)-groups are commensurable with fully invariant subgroups. This result is analogues to corresponding statement for direct sums of \(p\)-cyclic groups [\textit{B. Goldsmith} et al., J. Algebra 419, 332--349 (2014; Zbl 1305.20063)].
Reviewer: Nikolay I. Kryuchkov (Ryazan)Certain numerical results in non-associative structures.https://www.zbmath.org/1452.110262021-02-12T15:23:00+00:00"Azizi, Behnam"https://www.zbmath.org/authors/?q=ai:azizi.behnam"Doostie, Hossein"https://www.zbmath.org/authors/?q=ai:doostie.hosseinSummary: The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer \(n\ge 2\), the \(n\)th-commutativity degree of a finite algebraic structure \(S\), denoted by \(P_n(S)\), is the probability that for chosen randomly two elements \(x\) and \(y\) of \(S\), the relator \(x^ny=yx^n\) holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the \(n\)th-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for \(n\)th-commutativity degree of these loops, we will obtain best upper bounds for this probability.Homological approximations for profinite and pro-\(p\) limit groups.https://www.zbmath.org/1452.200242021-02-12T15:23:00+00:00"Gutierrez, Jhoel S."https://www.zbmath.org/authors/?q=ai:gutierrez.jhoel-sThis paper deals with the profinite completion of limit groups and the pro-\(p\) analogues of limit groups. Now limit groups have received much attention over the past decade, especially since they played an important role in the solution of the Tarski problem on free groups. One way to define limit groups is as finitely generated subgroups of groups obtained from free groups of finite rank by finitely many extensions of centralisers.
The first main result of the paper generalises a result of \textit{P. Zalesskii} and \textit{T. Zapata} [Rev. Mat. Iberoam. 36, No. 1, 61--78 (2020; Zbl 07197354)] on the asymptotic behaviour of the dimensions of the homology groups over \(\mathbb{F}_p\) of open normal subgroups of the profinite completion of a limit group. Specifically, for \(G\) a limit group, \(p\) a prime number, and \(\{U_i\}_{i\ge 1}\) a descending chain of open normal subgroups of the profinite completion \(\hat{G}\) of \(G\) such that the \(p\)-cohomological dimension \(\operatorname{cd}_p(\bigcap_{i\ge 1} U_i)\) of their intersection is at most 2, the author proves that:
(1) the limit of \(\dim H_j(U_i,\mathbb{F}_p)/[\hat{G}:U_i]\) as \(i\) goes to infinity is zero for \(j\ge 3\);
(2) if \(\{[\hat{G}:U_i]\}_{i\ge 1}\) tends to infinity, then the limit of \(\dim (H_1(U_i,\mathbb{F}_p) - H_2(U_i,\mathbb{F}_p))/[\hat{G}:U_i]\) as \(i\) goes to infinity is \(-\chi_p(\hat{G})\), where \(\chi_p(\hat{G})\) is the \(p\)-Euler characteristic of \(\hat{G}\);
(3) if \(\bigcap_{i\ge 1}U_i=1\), then the limit of \(\dim H_2(U_i,\mathbb{F}_p)/[\hat{G}:U_i]\) as \(i\) goes to infinity is zero and the limit of \(\dim H_1(U_i,\mathbb{F}_p)/[\hat{G}:U_i]\) is \(-\chi_p(\hat{G})\), which is non-negative.
This result is the profinite version of the corresponding result of \textit{D. Kochloukova} and \textit{P. Zalesskii} [Math. Nachr. 288, No. 5--6, 604--618 (2015; Zbl 1326.20031)] for pro-\(p\) limit groups. The author also establishes the result of Kochloukova and Zalesskii over \(\mathbb{Q}_p\). As a consequence, the author proves that for \(G\) being a pro-\(p\) limit group and \(\{U_i\}_{i\ge 1}\) a descending chain of open subgroups of \(G\) with trivial intersection, then the limit of \(T(U_i)/[G:U_i]\) is zero as \(i\) goes to infinity, where \(T(U_i)\) is the minimal number of generators of the torsion group of \(U_i^{\text{ab}}\).
Further, the author proves that if \(G\) is a non-abelian limit group and \(N\ne 1\) is a finitely generated closed normal subgroup of \(\hat{G}\), then \([\hat{G} : N]<\infty\). The discrete version of this result was proven by \textit{M. R. Bridson} and \textit{J. Howie} [Math. Ann. 337, No. 2, 385--394 (2007; Zbl 1139.20037)].
Reviewer: Anitha Thillaisundaram (Düsseldorf)On the number of irreducible real-valued characters of a finite group.https://www.zbmath.org/1452.200052021-02-12T15:23:00+00:00"Hung, Nguyen Ngoc"https://www.zbmath.org/authors/?q=ai:hung.nguyen-ngoc"Schaeffer Fry, A. A."https://www.zbmath.org/authors/?q=ai:schaeffer-fry.amanda-a"Tong-Viet, Hung P."https://www.zbmath.org/authors/?q=ai:tong-viet.hung-p"Vinroot, C. Ryan"https://www.zbmath.org/authors/?q=ai:vinroot.c-ryanLet \(G\) be a finite group with at most \(k\) real-valued irreducible characters. In this paper, the authors show that there exists an integer-valued function of the variable \(k\) such that \(|G/\text{Sol}(G)|\leq f(k)\), where \(\text{Sol}(G)\) denotes the soluble radical of \(G\). In the case \(k=5\), they prove that \(G/\text{Sol}(G)\) is isomorphic to the trivial group, \(\text{SL}_3(2)\), \(\text{Alt}(5)\), \(\text{PSL}_2(8)\cdot 3\), or \(^2\text{B}_2(8)\cdot 3\).
Reviewer: Mahmood Robati Sajjad (Qazvin)\(2\)-\((v,k,1)\) designs admitting automorphism groups with socle \(^2F_4(q)\).https://www.zbmath.org/1452.050112021-02-12T15:23:00+00:00"Li, Shangzhao"https://www.zbmath.org/authors/?q=ai:li.shangzhao"Dar, Shaojun"https://www.zbmath.org/authors/?q=ai:dar.shaojun"Han, Guangguo"https://www.zbmath.org/authors/?q=ai:han.guangguoSummary: It is a large and demanding project for determining pairs \((\mathcal{D},G)\) in which \(\mathcal{D}\) is a \(2\)-\((v,k,1)\) design and \(G\) is a block-transitive group of automorphisms of \(\mathcal{D}\). Eighteen years ago the problem was essentially reduced to the case in which \(G\) is an almost simple group, that is, a group such that for some non-abelian finite simple group \(T\) we have \(T\unlhd G\le\Aut(T)\) by \textit{A. R. Camina} and \textit{C. E. Praeger} [Aequationes Math. 61, No. 3, 221--232 (2001; Zbl 1058.20002)].\ This paper continues the works of [loc cit.]. Let \(G\) act as a point-primitive par block-transitive automorphism group of \(2\)-\((v,k,1)\) designs \(\mathcal{D}\) with a Ree group \(^2F_4(q)\) socle, we prove that the parameter \(k>\left\lceil\sqrt{\frac{43q^2-50a}{20a}}\right\rceil\), where \(q=2^a\) for an integer \(a=2n+1\), \(n>0\).An important step for the computation of the HOMFLYPT skein module of the Lens spaces \(L(p, 1)\) via braids.https://www.zbmath.org/1452.570052021-02-12T15:23:00+00:00"Diamantis, Ioannis"https://www.zbmath.org/authors/?q=ai:diamantis.ioannis"Lambropoulou, Sofia"https://www.zbmath.org/authors/?q=ai:lambropoulou.sofiaIn this paper the authors build on their previously established braid theoretic approach of deriving the Homflypt skein module of the lens spaces \(L(p,1)\) from its counterpart for the solid torus. Starting from two different bases for the skein module of the solid torus \(\mathcal{S}(ST)\), two different spanning sets can be obtained \(\Lambda ^{\prime\mathrm{aug}}\) and \(\Lambda^{\mathrm{aug}}\). Here they focus on the latter spanning set and they show that in order to obtain the Homflypt skein module for \(L(p,1)\), it suffices to do the following. First, consider braid band moves only on the first moving strand of the elements in \(\Lambda ^{\mathrm{aug}}\). Then, impose the invariant \(X\), which is the analogue of the Homflypt polynomial in the case of Coxeter groups of type B, and solve the derived infinite system of equations.
Reviewer: Dimos Goundaroulis (Lausanne)An invitation to coherent groups.https://www.zbmath.org/1452.570192021-02-12T15:23:00+00:00"Wise, Daniel T."https://www.zbmath.org/authors/?q=ai:wise.daniel-tA group is \textit{coherent} if every finitely generated subgroup has a finite presentation. The motivation comes from low-dimensional topology, in particular free groups, surface groups and 3-manifold groups are coherent. The present paper is an extensive survey on coherent groups, finishing with a list of 41 problems and 10 pages of references, and confronting coherence with various other concepts in combinatorial and geometric group theory. ``A primary motivation for me stems
from the desire to develop a class of groups that are close in nature to the fundamental groups of 3-manifolds, and echo one of their most salient properties.''
``There is a tremendous desire in mathematics to understand and classify all possible forms of some family of objects, and our human condition makes 3-manifolds appear rather quickly on the list of priorities. Nonetheless, I don't think 3-manifolds are ``for'' something else in mathematics, and I am unconvinced about how frequently they appear as crucial objects outside their domain -- in contrast to surfaces which are comparitively ubiquitous in mathematics. Instead, 3-manifolds are a topological ultimate goal, and I likewise hope coherent groups are an algebraic telos, that might motivate significant and useful technic.''
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)Shabat polynomials and monodromy groups of trees uniquely determined by ramification type.https://www.zbmath.org/1452.110732021-02-12T15:23:00+00:00"Cameron, Naiomi"https://www.zbmath.org/authors/?q=ai:cameron.naiomi-t"Kemp, Mary"https://www.zbmath.org/authors/?q=ai:kemp.mary"Maslak, Susan"https://www.zbmath.org/authors/?q=ai:maslak.susan"Melamed, Gabrielle"https://www.zbmath.org/authors/?q=ai:melamed.gabrielle"Moy, Richard A."https://www.zbmath.org/authors/?q=ai:moy.richard-a"Pham, Jonathan"https://www.zbmath.org/authors/?q=ai:pham.jonathan"Wei, Austin"https://www.zbmath.org/authors/?q=ai:wei.austinA tree is a graph without any loops. A plane tree is a tree with an embedding into the plane or equivalently a tree together with an ordering of edges emanating from vertices. A map of plane trees preserving this extra structure is called an isomorphism of plane trees. Enumeration problem of isomorphism classes of plane trees as well as their complete list (in terms of their ramification types) is addressed in [\textit{G. Shabat} and \textit{A. Zvonkin}, Contemp. Math. 178, 233--275 (1994; Zbl 0816.05024)].
Building on this, the authors determine the corresponding Belyi maps (called Shabat polynomials) and their monodromy groups.
Reviewer: Ayberk Zeytin (Istanbul)The congruence subgroup problem for a family of branch groups.https://www.zbmath.org/1452.200232021-02-12T15:23:00+00:00"Skipper, Rachel"https://www.zbmath.org/authors/?q=ai:skipper.rachelLet \(T\) be a rooted tree, and \(G\) be a subgroup of \(\text{Aut}(T)\). For \(n\in\mathbb{N}\), we write \(\text{St}_G(n)\) for the pointwise stabiliser of the \(n\)th level of the tree \(T\). The group \(G\) is said to have the congruence subgroup property if every subgroup of fiinite index contains \(\text{St}_G(n)\) for some \(n\in\mathbb{N}\). Equivalently, the group \(G\) has the congruence subgroup property if the topological closure \(\overline{G}\) of \(G\) in \(\text{Aut}(T)\) equals the profinite completion \(\widehat{G}\) of \(G\). The congruence subgroup problem then asks to determine the kernel of the natural surjection from \(\widehat{G}\) to \(\overline{G}\). This kernel is called the congruence kernel.
The study of congruence kernels in branch groups was initiated by \textit{L. Bartholdi} et al. [Isr. J. Math. 187, 419--450 (2012; Zbl 1271.20031)], and they split the determination of the congruence kernel into two parts: the branch kernel and the rigid kernel.
Many known examples of branch groups have trivial congruence kernel, and most examples of branch groups with non-trivial congruence kernel have trivial rigid kernel. Prior to this paper, the only example of a branch group with non-trivial rigid kernel was the Hanoi towers group acting on the 3-adic tree. By considering subgroups of a natural generalisation of the Hanoi towers group acting on the \(n\)-adic tree, for \(n\ge 4\), the author constructs infinitely many branch groups with non-trivial rigid kernel. Additionally, it is shown that these generalised Hanoi towers groups themselves have trivial rigid kernel and information on the form of their congruence kernels is given.
Also in this paper, further results on these generalised Hanoi towers groups \(G_n\) are given, such as that the group \(G_n\) is just infinite if and only if \(n\ne 3\), and it is shown that the Hausdorff dimension of the closure \(\overline{G_n}\) approaches 1 as \(n\) goes to infinity.
Reviewer: Anitha Thillaisundaram (Düsseldorf)Construction of subsurfaces via good pants.https://www.zbmath.org/1452.570012021-02-12T15:23:00+00:00"Liu, Yi"https://www.zbmath.org/authors/?q=ai:liu.yi"Markovic, Vladimir"https://www.zbmath.org/authors/?q=ai:markovic.vladimirThe good pants technology is a systematic method to produce surface subgroups in cocompact lattices of \(\mathrm{PSL}_2(\mathbb C)\) and \(\mathrm{PSL}_2(\mathbb R)\) (using frame flow). The \(\mathrm{PSL}_2(\mathbb C)\)-case leads to a proof of the Surface Subgroup Conjecture [\textit{J. Kahn} and \textit{V. Markovic}, Ann. Math. (2) 175, No. 3, 1127--1190 (2012; Zbl 1254.57014)], and the \(\mathrm{PSL}_2(\mathbb R)\)-case to a proof of the Ehrenpreis Conjecture on Riemann surfaces [\textit{J. Kahn} and \textit{V. Markovic}, in: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13--21, 2014. Vol. II: Invited lectures. Seoul: KM Kyung Moon Sa. 897--909 (2014; Zbl 1373.57037)].
The \textit{Surface Subgroup Conjecture} asserts that any cocompact lattice in \(\mathrm{PSL}_2(\mathbb C)\) contains a surface subgroup, in particular any closed hyperbolic 3-manifold contains \(\pi_1\)-injective immersed subsurfaces (this plays a fundamental role in the resolution of the \textit{Virtual Haken Conjecture} for hyperbolic 3-manifolds).
The \textit{Ehrenpreis Conjecture} asserts that, for any two closed Riemann surfaces of the same genus and positive constant \(\epsilon > 0\), there exists a \((1+ \epsilon)\)-quasiconformal map between some finite covers of the two surfaces; equivalently, any two closed Riemann surfaces are virtually \((1+ \epsilon)\)-bilipschitz equivalent, for any positive constant \(\epsilon\).
The present paper is a survey on the constructions and methods leading to the solutions of these two conjectures, and also on some further applications of the methods.
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)Some applications of algebraic entropy to the proof of Milnor-Wolf theorem.https://www.zbmath.org/1452.200392021-02-12T15:23:00+00:00"Xi, W."https://www.zbmath.org/authors/?q=ai:xi.wenfei"Dikranjan, D."https://www.zbmath.org/authors/?q=ai:dikranjan.dikran-n"Freni, D."https://www.zbmath.org/authors/?q=ai:freni.domenico"Toller, D."https://www.zbmath.org/authors/?q=ai:toller.danieleSeveral nice results are given on the algebraic entropy \(h_{alg}(\phi)\) and the growth of a group endomorphism \(\phi\colon G\to G\). These results are applied to give a new proof of the classical Milnor-Wolf theorem, that is, of the fact that there exist no finitely generated solvable group of intermediate growth. More precisely, the authors show that a finitely generated solvable group \(G\) of subexponential growth is virtually nilpotent, and then the Gromov theorem gives that equivalently \(G\) has polynomial growth.
To achieve this goal, the authors study the growth of a finitely generated cascade, namely, a pair \((G,\phi)\) where \(G\) is a group, \(\phi\colon G\to G\) is an automorphism and there exists a finite subset \(F\) of \(G\) such that \(G=\langle\phi^n(F)\colon n\in\mathbb N\rangle\).
The growth of a finitely generated cascade \((G,\phi)\) is related to the classical growth of the semidirect product \(G\rtimes \langle\phi\rangle\). For example, the group \(G\rtimes \langle\phi\rangle\) is finitely generated if and only if the cascade \((G,\phi)\) is finitely generated, and if \(G\rtimes \langle\phi\rangle\) has subexponential growth then \(G\) is finitely generated as well; in order to obtain the latter property the authors apply their theorem stating that in case \(h_{\mathrm{alg}}(\phi)<\log 2\), then \(G\) is finitely generated. Moreover, if \(G\) is nilpotent, then the group \(G\rtimes \langle\phi\rangle\) has the same growth type of the cascade \((G,\phi)\), and this growth type cannot be intermediate.
Reviewer: Anna Giordano Bruno (Udine)The ABC of \(p\)-cells.https://www.zbmath.org/1452.200492021-02-12T15:23:00+00:00"Jensen, Lars Thorge"https://www.zbmath.org/authors/?q=ai:jensen.lars-thorgeThe paper is dedicated to developing a positive characteristic version of the Kazhdan-Lusztig cells. In [Contemp. Math. 683, 333--361 (2017; Zbl 1390.20001)], the author and \textit{G. Williamson} defined the \(p\)-canonical basis for the Hecke algebra of a crystallographic Coxeter system. It can be thought of as a positive characteristic analogue of the Kazhdan-Lusztig basis. The \(p\)-canonical basis shares strong positivity properties with the Kazhdan-Lusztig basis (similar to the ones described by the Kazhdan-Lusztig positivity conjectures), but it loses many of its combinatorial properties. Replacing the Kazhdan-Lusztig basis by the \(p\)-canonical basis in the definition of the left (resp. right or two-sided) cells leads to the notion of left (resp. right or two-sided) p-cells. These \(p\)-cells are the main subject of the present paper.
The first properties of \(p\)-cells are proved in Section 3. Left and right \(p\)-cells are related by taking inverses (see Lemma 3.6), just like for Kazhdan-Lusztig cells. The set of elements with a fixed left descent set decomposes into right \(p\)-cells (see Lemma 3.4). The most important result of this section is a certain compatibility of \(p\)-cells with parabolic subgroups that shows that any right \(p\)-cell preorder relation in a finite standard parabolic subgroup \(W_I\) induces right \(p\)-cell preorder relations in each right \(W_I\)-coset (see Theorem 3.9). It is also shown that unfortunately Kazhdan-Lusztig cells do not always decompose into \(p\)-cells, but the author expects that this may still be the case when the prime \(p\) is good for the corresponding algebraic group. Indeed, in Section 4 it is proved that Kazhdan-Lusztig left cells decompose into left \(p\)-cells in finite types B and C for \(p > 2\). This is done by studying the consequences of the Kazhdan-Lusztig star-operations for the \(p\)-canonical basis, these operations appear as the main technical tool of the present work.
Reviewer: Mikhail Belolipetsky (Rio de Janeiro)