Recent zbMATH articles in MSC 20Fhttps://www.zbmath.org/atom/cc/20F2022-05-16T20:40:13.078697ZWerkzeugOn simple connectivity of random 2-complexeshttps://www.zbmath.org/1483.052052022-05-16T20:40:13.078697Z"Luria, Zur"https://www.zbmath.org/authors/?q=ai:luria.zur"Peled, Yuval"https://www.zbmath.org/authors/?q=ai:peled.yuvalSummary: The fundamental group of the 2-dimensional Linial-Meshulam random simplicial complex \(Y_2(n,p)\) was first studied by \textit{E. Babson} et al. [J. Am. Math. Soc. 24, No. 1, 1--28 (2011; Zbl 1270.20042)]. They proved that the threshold probability for simple connectivity of \(Y_2(n,p)\) is about \(p\approx n^{-1/2}\). In this paper, we show that this threshold probability is at most \(p\le (\gamma n)^{-1/2}\), where \(\gamma =4^4/3^3\), and conjecture that this threshold is sharp. In fact, we show that \(p=(\gamma n)^{-1/2}\) is a sharp threshold probability for the stronger property that every cycle of length 3 is the boundary of a subcomplex of \(Y_2(n,p)\) that is homeomorphic to a disk. Our proof uses the Poisson paradigm, and relies on a classical result of Tutte on the enumeration of planar triangulations.Koszul algebras and quadratic duals in Galois cohomologyhttps://www.zbmath.org/1483.120032022-05-16T20:40:13.078697Z"Mináč, Jan"https://www.zbmath.org/authors/?q=ai:minac.jan"Pasini, Federico William"https://www.zbmath.org/authors/?q=ai:pasini.federico-william"Quadrelli, Claudio"https://www.zbmath.org/authors/?q=ai:quadrelli.claudio"Nguyễn Duy Tân"https://www.zbmath.org/authors/?q=ai:nguyen-duy-tan.Given a field \(F\) and a prime number \(p\), the paper under review investigates the Galois cohomology of the Galois group \(\mathcal{G}(F(p)/F)\) of the maximal \(p\)-extension \(F(p)/F)\), under the condition that \(\mathcal{G}(F(p)/F)\) is finitely-generated as a pro-\(p\)-group; by Galois theory, \(\mathcal{G}(F(p)/F)\) is isomorphic as a profinite group to the maximal pro-\(p\) quotient of the absolute Galois group \(\mathcal{G}_F := \mathcal{G}(F_{\text{sep}}/F)\), where \(F_{\text{sep}}\) is a separable closure of \(F\). It presents a research motivated by the open problem of finding which profinite groups are realizable as absolute Galois groups. The Galois cohomology of \(F\) with coefficients in the prime field \(\mathbb{F}_F := \mathbb{Z}/p\mathbb{Z}\) is a graded algebra \(H^*(\mathcal{G}_F, \mathbb{F}_p) = \oplus_{n \ge 0} H^n(\mathcal{G}_F, \mathbb{F}_p)\) with respect to the homological degree and the graded-commutative cup product \(\cup: H^r(\mathcal{G}_F, \mathbb{F}_p) \times H^s(\mathcal{G}_F, \mathbb{F}_p) \to H^{r+s}(\mathcal{G}_F, \mathbb{F}_p)\), \(r, s \ge 0\) (see Ch. 1, Sect. 4 of the book [\textit{J. Neukirch} et al., Cohomology of number fields. Berlin: Springer (2008; Zbl 1136.11001)]).
Throughout the reviewed paper, the authors assume that \(F\) contains a primitive \(p\)-th root of unity \(\xi_p\), and in case \(p = 2\), \(\sqrt{-1} \in F\). Their research has been inspired by the proof of the Bloch-Kato conjecture (completed by Voevodsky and Rost, see [\textit{A. Suslin} and \textit{S. Joukhovitski}, J. Pure Appl. Algebra 206, No. 1--2, 245--276 (2006; Zbl 1091.19002); \textit{C. Haesemeyer} and \textit{C. Weibel}, Abel Symp. 4, 95--130 (2009; Zbl 1244.19003); \textit{V. Voevodsky}, Publ. Math., Inst. Hautes Étud. Sci. 112, 1--99 (2010; Zbl 1227.14025); \textit{V. Voevodsky}, Ann. Math. (2) 174, No. 1, 401--438 (2011; Zbl 1236.14026)]), which implies that \(H^*(\mathcal{G}_F, \mathbb{F}_p)\), i.e., it is generated by elements of degree \(1\) and its relations are generated by homogeneous relations of degree \(2\).
It is known that locally finite-dimensional quadratic algebras come equipped with a duality (see Page 1 of the lectures [\textit{A. Polishchuk} and \textit{L. Positselski}, Trans. Am. Math. Soc. 364, No. 10, 5311--5368 (2012; Zbl 1285.16005)]. The quadratic dual of such an algebra A\(\bullet\) is a quadratic algebra generated over the same field by the dual space of the space of generators of A\(\bullet\). Its relators form the orthogonal complement of the space of the relators of A\(\bullet\) (see Definition 2.3). The double quadratic dual (A!\(\bullet\))! is isomorphic to A\(\bullet\) as a graded algebra. Among quadratic algebras, the significant class of Koszul algebras has been introduced in [\textit{S. B. Priddy}, Trans. Am. Math. Soc. 152, 39--60 (1970; Zbl 0261.18016)] as well as characterized by their uncommonly nice cohomological properties. In this respect, the Bloch-Kato conjecture is strengthened by Positselski as follows: \par Conjecture. If \(F\) is a field containing a primitive \(p\)-th root of unity, then \(H \sp *(\mathcal{G}_F, \mathbb{F}_p)\) is a Koszul algebra.
In order that A\(\bullet\) be a Koszul algebra, it is sufficient (but not necessary) that A\(\bullet\) possesses the PBW (Poincaré-Birkhof-Witt) property. Therefore, it should be noted that, generally, it is easier to check the PBW property than to verify the Koszul property. In addition, the paper under review proves Positselski's conjecture in a number of special cases, by showing that: in each considered case, \(H \sp *(\mathcal{G} \sb F, \mathbb{F} \sb p)\) has the PBW property; under the above-noted restrictions on the roots of unity in \(F\), \(H^*(\mathcal{G}_F, \mathbb{F}_p)\) has the PBW property, provided that \(\mathcal{G}(F(p)/F)\) is an elementary type pro-\(p\)-group (in view of the Elementary Type Conjecture, this occurs in all presently known situations); an analogous unconditional result about Pythagorean fields is obtained as well. In particular, the availability of the PBW property is proved in the following cases: (a) \(F\) is finite; (b) \(\mathcal{G}(F(p)/F)\) is a Demushkin pro-\(p\)-group; (c) \(F\) is a pseudo algebraically closed (PAC) field, or an extension of transcendence degree 1 of a PAC field; (c) \(F\) is an algebraic extension of \(\mathbb{Q}\); (d) \(F\) is a local field or an extension of transcendence degree \(1\) of a local field; (e) \(F\) is \(p\)-rigid.
The paper also contains a survey of the Koszul property in Galois cohomology and its relation with absolute Galois groups.
Reviewer: Ivan D. Chipchakov (Sofia)Bases of the intersection cohomology of Grassmannian Schubert varietieshttps://www.zbmath.org/1483.140952022-05-16T20:40:13.078697Z"Patimo, Leonardo"https://www.zbmath.org/authors/?q=ai:patimo.leonardoThe main goal of the present paper is to lift the combinatorics of Dyck partitions to the category of Grassmannian Soergel bimodules, obtaining a basis of the intersection cohomology parameterized by Dyck partitions.
The basic idea is to reinterpret each Dyck strip \(D\) as a morphism of degree one (denoted by \(f_D\)) between the corresponding singular Soergel bimodules. A key feature of indecomposable Grassmannian Soergel bimodules is that the space of degree one morphisms between indecomposable bimodules is one-dimensional, therefore the morphism \(f_D\) is actually uniquely determined up to a scalar.
For an arbitrary Dyck partition \(P = \{D_1, D_2,\dots, D_k\}\), they consider the morphism \(f_P = f_{D_1} \circ f{_D{_2}} \circ\dots\circ f_{D_k}\). Unfortunately, this morphism is not well defined: different orders of the elements in \(P\) (i.e. different orders for the composition of the morphisms \(f_{D_i}\)) may lead to different morphisms. A crucial technical point is to define a partial order on the set of Dyck partitions. They then show that the morphism \(f_P\) is well defined, up to a scalar and up to smaller morphisms in the partial order.
After fixing for any Dyck partition an order of its strips arbitrarily the set \(\{f_P\}\) gives us a basis of the morphisms between singular Soergel bimodules. Moreover, the basis they obtain is cellular and this makes the Grassmannian Soergel bimodules a strictly object-adapted cellular category. By evaluating these morphisms on the unit of the cohomology ring, they also get bases of the (equivariant) intersection cohomology of Schubert varieties.
Reviewer: Cenap Özel (İzmir)The classification of blocks in BGG category \(\mathcal{O}\)https://www.zbmath.org/1483.170092022-05-16T20:40:13.078697Z"Coulembier, Kevin"https://www.zbmath.org/authors/?q=ai:coulembier.kevinA block in the BGG category \(\mathcal{O}\) of a complex semisimple Lie algebra is determined up to equivalence by its integral Weyl group \(W\) and a parabolic subgroup \(W^\prime \leq W\) by results of [\textit{W. Soergel}, J. Am. Math. Soc. 3, No. 2, 421--445 (1990; Zbl 0747.17008)]. Therefore, a block can be denoted unambigously by \(\mathcal{O}(W,W^\prime)\).
The main result of this article is that, for any two finite Weyl groups \(W\) and \(U\) with parabolic subgroups \(W^\prime \leq W\) and \(U^\prime \leq U\), the blocks \(\mathcal{O}(W,W^\prime)\) and \(\mathcal{O}(U,U^\prime)\) are equivalent if and only if the Bruhat orders on \(W / W^\prime\) and \(U / U^\prime\) are isomorphic. As part of the proof, it is observed that any finite-dimensional algebra with simple preserving duality admits at most one quasi-hereditary structure. The author further determines all pairs \((W,W^\prime)\) and \((U,U^\prime)\) such that there exists an isomorphisms between the Bruhat orders on \(W / W^\prime\) and \(U / U^\prime\) and thus obtains a complete classification of the blocks in category \(\mathcal{O}\).
Reviewer: Jonathan Gruber (Lausanne)The B B Newman spelling theoremhttps://www.zbmath.org/1483.200012022-05-16T20:40:13.078697Z"Nyberg-Brodda, Carl-Fredrik"https://www.zbmath.org/authors/?q=ai:nyberg-brodda.carl-fredrikThis entertaining article is about the history of the B. B. Newman spelling theorem, a crucial result in combinatorial group theory. The paper provides a self-contained account of the growth of combinatorial group theory as a research field from late 19th century, and summarises its main problems and results. In the first section of the article, the author recounts the story of the person behind the theorem, namely the Australian mathematician Bill Bateup Newman. With the aid of the mathematicians directly involved in B. B. Newman's early career, of B. B. Newman himself, as well as the librarians and archivists of the James Cook University and Queensland University in Australia, the author was able to track down B. B. Newman's PhD thesis and establish that he was officially supervised by the American-based mathematician Gilbert Baumslag. Thanks to the author's effort, Newman's thesis is digitised and now available online at the James Cook University's library website [\textit{B. B. Newman}, Some aspects of one-relator group. Townsville City: University College of Townsville (PhD Thesis) (1968; \url{doi:10.25903/5cb6b32e38c08})]. Particularly noteworthy is the international dimension of collaboration between mathematicians behind B. B. Newman's interest for the subject of combinatorial group theory and his doctoral dissertation.
Reviewer: Davide Crippa (Praha)On the universal \(\mathrm{SL}_2\)-representation rings of free groupshttps://www.zbmath.org/1483.200072022-05-16T20:40:13.078697Z"Satoh, Takao"https://www.zbmath.org/authors/?q=ai:satoh.takaoSummary: In this paper, we give an explicit realization of the universal \(\mathrm{SL}_2\)-representation rings of free groups by using `the ring of component functions' of \(\mathrm{SL}(2, \mathbb{C})\)-representations of free groups. We introduce a descending filtration of the ring, and determine the structure of its graded quotients. Then we study the natural action of the automorphism group of a free group on the graded quotients, and introduce a generalized Johnson homomorphism. In the latter part of this paper, we investigate some properties of these homomorphisms from a viewpoint of twisted cohomologies of the automorphism group of a free group.Young tableaux and representations of Hecke algebras of type ADEhttps://www.zbmath.org/1483.200082022-05-16T20:40:13.078697Z"Poulain d'Andecy, Loïc"https://www.zbmath.org/authors/?q=ai:poulain-dandecy.loicSummary: We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of type ADE. Representations are constructed by explicit formulas, in a seminormal form.On zeros of characters of finite groups and solvable \( \phi \)-groupshttps://www.zbmath.org/1483.200152022-05-16T20:40:13.078697Z"Ren, Yongcai"https://www.zbmath.org/authors/?q=ai:ren.yongcai"Zhang, Jinshan"https://www.zbmath.org/authors/?q=ai:zhang.jinshan.1|zhang.jinshan(no abstract)On Schur multiplier and projective representations of Heisenberg groupshttps://www.zbmath.org/1483.200272022-05-16T20:40:13.078697Z"Hatui, Sumana"https://www.zbmath.org/authors/?q=ai:hatui.sumana"Singla, Pooja"https://www.zbmath.org/authors/?q=ai:singla.pooja|singla.pooja.1The theory of projective representations involves understanding homomorphisms from a group into the projective linear groups. By definition, every ordinary representation of a group is projective, but the converse is not true.
The Schur multiplier of a group \(G\) is the second cohomology group \(\mathrm{H}^2(G, \mathbb{C}^{\times})\), where \(\mathbb{C}^{\times}\) is a trivial \(G\)-module and, for that, it is a useful tool to understanding projective representations.
In the paper under review, the authors describe the Schur multiplier, the representation of discrete Heisenberg groups and their \(t\)-variants. Furthermore, they provide a construction of all complex finite dimensional irreducible projective representations of these groups.
Reviewer: Enrico Jabara (Venezia)Fibers of word maps and the multiplicities of non-abelian composition factorshttps://www.zbmath.org/1483.200302022-05-16T20:40:13.078697Z"Bors, Alexander"https://www.zbmath.org/authors/?q=ai:bors.alexanderOrders of automorphism groups of some groups of order \(p^6\)https://www.zbmath.org/1483.200442022-05-16T20:40:13.078697Z"Su, Hong"https://www.zbmath.org/authors/?q=ai:su.hong"Zhao, Zhen Hua"https://www.zbmath.org/authors/?q=ai:zhao.zhenhua"Qiu, Dun Yuan"https://www.zbmath.org/authors/?q=ai:qiu.dunyuan(no abstract)Orders of the automorphism groups of some \(p\)-groupshttps://www.zbmath.org/1483.200462022-05-16T20:40:13.078697Z"Wang, Yong"https://www.zbmath.org/authors/?q=ai:wang.yong.10|wang.yong.9|wang.yong|wang.yong.11|wang.yong.5|wang.yong.2|wang.yong.3|wang.yong.8|wang.yong.1|wang.yong.7"Ban, Gui Ning"https://www.zbmath.org/authors/?q=ai:ban.guining(no abstract)The lamplighter group of rank two generated by a bireversible automatonhttps://www.zbmath.org/1483.200492022-05-16T20:40:13.078697Z"Ahmed, Elsayed"https://www.zbmath.org/authors/?q=ai:ahmed.elsayed-a"Savchuk, Dmytro"https://www.zbmath.org/authors/?q=ai:savchuk.dmytro-mConsider a deterministic, complete, finite state automaton \(\mathcal A\) on an alphabet \(\Sigma\) with letter-to-letter transducer. If one fixes an initial state, one obtains a length preserving transformation on the set \(\Sigma^*\) of words on the alphabet \(\Sigma\), which can be viewed as a \(|\Sigma|\)-regular rooted tree \(\mathcal T_\Sigma\).
Varying the initial state, one gets finitely many such transformations. When they are all bijective, there is a canonically defined automaton \(\mathcal A^{-1}\) whose associated transformations are the inverses of the transformations associated to \(\mathcal A\). The group of transformations of \(\Sigma^*\) they generate is called an automaton group. Grigorchuk's group of intermediate growth is a prominent example of such a group.
The authors construct an explicit 4-state 2-letter automaton \(\mathcal A\) such that the four associated transformations of the regular rooted binary tree generate a lamplighter group \((\mathbb{Z}_2^2)\wr \mathbb{Z}\) of rank two. Moreover, the automaton is \emph{bi-reversible}: this means that the dual automata (obtained by reversing the roles of the set of states and of the alphabet) of \(\mathcal A\) and \(\mathcal A^{-1}\) are both invertible (i.e.\ induce bijections of the associated rooted trees).
Additionally, using the natural identification of the boundary of \(\mathcal T_\Sigma\) with the ring of formal series \(\mathbb Z_2[[t]]\), the authors show that the automaton group is actually contained in its group of affine transformations \(\mathbb Z_2[[t]]^{\times}\ltimes \mathbb Z_2[[t]]\).
Reviewer: Vincent Guirardel (Rennes)Compact presentability of tree almost automorphism groupshttps://www.zbmath.org/1483.200502022-05-16T20:40:13.078697Z"Le Boudec, Adrien"https://www.zbmath.org/authors/?q=ai:le-boudec.adrienSummary: We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results include Neretin's group of spheromorphisms, as well as the topologically simple group containing the profinite completion of the Grigorchuk group constructed by Barnea, Ershov and Weigel.
We additionally obtain an upper bound on the Dehn function of these groups in terms of the Dehn function of an embedded Higman-Thompson group. This, combined with a result of Guba, implies that the Dehn function of the Neretin group of the regular trivalent tree is polynomially bounded.Characterization of finitely generated groups by typeshttps://www.zbmath.org/1483.200552022-05-16T20:40:13.078697Z"Myasnikov, A. G."https://www.zbmath.org/authors/?q=ai:myasnikov.alexei-g"Romanovskii, N. S."https://www.zbmath.org/authors/?q=ai:romanovskii.n-sOn groups with finite conjugacy classes in a verbal subgrouphttps://www.zbmath.org/1483.200572022-05-16T20:40:13.078697Z"Delizia, Costantino"https://www.zbmath.org/authors/?q=ai:delizia.costantino"Shumyatsky, Pavel"https://www.zbmath.org/authors/?q=ai:shumyatsky.pavel"Tortora, Antonio"https://www.zbmath.org/authors/?q=ai:tortora.antonioSummary: Let \(w\) be a group-word. For a group \(G\), let \(G_{w}\) denote the set of all \(w\)-values in \(G\) and let \(w(G)\) denote the verbal subgroup of \(G\) corresponding to \(w\). The group \(G\) is an \(FC(w)\)-group if the set of conjugates \(x^{G_{w}}\) is finite for all \(x\in G\). It is known that if \(w\) is a concise word, then \(G\) is an \(FC(w)\)-group if and only if \(w(G)\) is \(FC\)-embedded in \(G\), that is, the conjugacy class \(x^{w(G)}\) is finite for all \(x\in G\). There are examples showing that this is no longer true if \(w\) is not concise. In the present paper, for an arbitrary word \(w\), we show that if \(G\) is an \(FC(w)\)-group, then the commutator subgroup \(w(G)^{\prime}\) is \(FC\)-embedded in \(G\). We also establish the analogous result for \(BFC(w)\)-groups, that is, groups in which the sets \(x^{G_{w}}\) are boundedly finite.Finite groups of arbitrary deficiencyhttps://www.zbmath.org/1483.200582022-05-16T20:40:13.078697Z"Gardam, Giles"https://www.zbmath.org/authors/?q=ai:gardam.gilesSummary: The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. We show that every non-positive integer is the deficiency of a finite group -- in fact, of a finite \(p\)-group for every prime \(p\). This completes \textit{D. Kotschick}'s classification [J. Topol. 5, No. 3, 639--650 (2012; Zbl 1256.20033)] of the integers which are deficiencies of fundamental groups of compact Kähler manifolds.Infinitely presented graphical small cancellation groups are acylindrically hyperbolichttps://www.zbmath.org/1483.200592022-05-16T20:40:13.078697Z"Gruber, Dominik"https://www.zbmath.org/authors/?q=ai:gruber.dominik"Sisto, Alessandro"https://www.zbmath.org/authors/?q=ai:sisto.alessandroSummary: We prove that infinitely presented graphical \(Gr(7)\) small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical \(C(7)\)-groups and, hence, classical \(C'(\frac{1}{6})\)-groups are acylindrically hyperbolic. We also prove the analogous statements for the larger class of graphical small cancellation presentations over free products. We construct infinitely presented classical \(C'(\frac{1}{6})\)-groups that provide new examples of divergence functions of groups.Computability of Følner setshttps://www.zbmath.org/1483.200602022-05-16T20:40:13.078697Z"Cavaleri, Matteo"https://www.zbmath.org/authors/?q=ai:cavaleri.matteoObstruction to a Higman embedding theorem for residually finite groups with solvable word problemhttps://www.zbmath.org/1483.200612022-05-16T20:40:13.078697Z"Rauzy, Emmanuel"https://www.zbmath.org/authors/?q=ai:rauzy.emmanuelThe well known Higman's embedding theorem states that every finitely generated recursively presented group can be embedded as a subgroup of some finitely presented group. On the other hand, not all finitely generated recursively presented residually finite groups embed into finitely presented residually finite groups. On the basis of the following observations
\begin{itemize}
\item[1.] all finitely presented residually finite groups have a solvable word problem;
\item[2.] there are many recursively presented residually finite groups with a non-solvable word problem;
\end{itemize}
\textit{O. Kharlampovich} et al. [Bull. Math. Sci. 7, No. 2, 309--352 (2017; Zbl 1423.20022)] deduced that the non-existence of the embedding could have been related only to the non-solvability of the word problem. The aim of this paper is to show that this is not actually the case. In fact, Theorem 1 shows that there exist finitely generated recursively presented residually finite groups with solvable word problem that do not embed into any finitely presented residually finite group. In doing so, the author make a strong use of \textit{Dyson's groups}, i.e. amalgamated products of two lamplighter groups, to reduce the proof to a problem about subsets of \(\mathbb{Z}\).
Reviewer: Maria Ferrara (Caserta)Nilpotency in uncountable groupshttps://www.zbmath.org/1483.200622022-05-16T20:40:13.078697Z"de Giovanni, Francesco"https://www.zbmath.org/authors/?q=ai:de-giovanni.francesco"Trombetti, Marco"https://www.zbmath.org/authors/?q=ai:trombetti.marcoSummary: The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality \(\aleph\) in which all proper subgroups of cardinality \(\aleph\) are nilpotent. It is proved that such a group \(G\) is nilpotent, provided that \(G\) has no infinite simple homomorphic images and either \(\aleph\) has cofinality strictly larger than \(\aleph_{0}\) or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.The \(q\)-tensor square of finitely generated nilpotent groups, \(q\) oddhttps://www.zbmath.org/1483.200632022-05-16T20:40:13.078697Z"Rocco, Noraí R."https://www.zbmath.org/authors/?q=ai:rocco.norai-romeu"Rodrigues, Eunice C. P."https://www.zbmath.org/authors/?q=ai:rodrigues.eunice-candida-pereiraSome nilpotency criteria for groupshttps://www.zbmath.org/1483.200642022-05-16T20:40:13.078697Z"Wang, Yu Lei"https://www.zbmath.org/authors/?q=ai:wang.yulei"Liu, He Guo"https://www.zbmath.org/authors/?q=ai:liu.heguo(no abstract)On groups with all proper subgroups finite-by-abelian-by-finitehttps://www.zbmath.org/1483.200652022-05-16T20:40:13.078697Z"Dardano, Ulderico"https://www.zbmath.org/authors/?q=ai:dardano.ulderico"De Mari, Fausto"https://www.zbmath.org/authors/?q=ai:de-mari.faustoIf \(\mathcal{P}\) is a class of groups it is possible to define a new class \(\overline{\mathcal{P}}\) consisting of all those groups in which every proper subgroup is a \(\mathcal{P}\)-group. Let \(\mathcal{P}=\mathtt{FAF}\) the property of being finite by-abelian-by-finite, hence a group \(G\) is a \(\mathcal{P}\)-group if \(G\) has a subgroup \(K\) with finite index such that the derived subgroup \(K\) is finite. The aim of the paper under review is to study \(\overline{\mathcal{P}}\)-groups. The main result proved in this article is Theorem A: Let \(G\) be a locally graded group. If \(G \in \overline{\mathcal{P}}\), then \(G\) contains a finite normal subgroup \(N\) such that all proper subgroups of \(G/N\) are abelian-by-finite. In particular, if \(G\) is minimal-non-(finite-by-abelian-by-finite), then \(G\) is periodic and finite-by-metabelian.
A group \(G\) if a \(\mathtt{CF}\)-group if each subgroup \(H\) of \(G\) is core-finite, i.e. \(H/H_{G}\) is finite. A group \(G\) is a \(\mathtt{CN}\)-group if each subgroup of \(G\) is commensurable to a normal subgroup. Another interesting result is Theorem B: Let \(G\) be a locally graded group whose periodic sections are locally finite. If all proper subgroups of \(G\) are \(\mathtt{CN}\)-groups, then \(G\) contains a finite normal subgroup \(N\) such that all proper subgroups of \(G/N\) are \(\mathtt{CF}\)-groups, hence \(G\) is finite-by-abelian-by-finite.
Reviewer: Enrico Jabara (Venezia)Infinite groups acting faithfully on the outer automorphism group of a right-angled Artin grouphttps://www.zbmath.org/1483.200662022-05-16T20:40:13.078697Z"Bregman, Corey"https://www.zbmath.org/authors/?q=ai:bregman.corey"Fullarton, Neil J."https://www.zbmath.org/authors/?q=ai:fullarton.neil-jSummary: We construct the first known examples of infinite subgroups of the outer automorphism group of \(\mathrm{Out}(A_\Gamma)\), for certain right-angled Artin groups \(A_\Gamma\). This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of \(\mathrm{Out}(\mathrm{Out}(A_\Gamma))\). This demonstrates a marked departure from the known behavior of \(\mathrm{Out}(\mathrm{Out}(A_\Gamma))\) when \(A_\Gamma\) is free or free abelian since in these cases \(\mathrm{Out}(\mathrm{Out}(A_\Gamma))\) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite-order members of certain \(\mathrm{Out}(\Aut(A_\Gamma))\).The stable automorphism group of a given semidirect producthttps://www.zbmath.org/1483.200672022-05-16T20:40:13.078697Z"Zhou, Fang"https://www.zbmath.org/authors/?q=ai:zhou.fang"Ma, Yu Jie"https://www.zbmath.org/authors/?q=ai:ma.yujie"Liu, He Guo"https://www.zbmath.org/authors/?q=ai:liu.heguo(no abstract)Artin groups of infinite type: trivial centers and acylindrical hyperbolicityhttps://www.zbmath.org/1483.200682022-05-16T20:40:13.078697Z"Charney, Ruth"https://www.zbmath.org/authors/?q=ai:charney.ruth-m"Morris-Wright, Rose"https://www.zbmath.org/authors/?q=ai:morris-wright.roseSummary: While finite type Artin groups and right-angled Artin groups are well understood, little is known about more general Artin groups. In this paper, we use the action of an infinite type Artin group \(A_{\Gamma}\) on a CAT(0) cube complex to prove that \(A_{\Gamma}\) has trivial center providing \(\Gamma\) is not the star of a single vertex, and is acylindrically hyperbolic providing \(\Gamma\) is not a join.A product on double cosets of \(B_\infty\)https://www.zbmath.org/1483.200692022-05-16T20:40:13.078697Z"Gonzalez Pagotto, Pablo"https://www.zbmath.org/authors/?q=ai:gonzalez-pagotto.pabloSummary: For some infinite-dimensional groups \(G\) and suitable subgroups \(K\) there exists a monoid structure on the set \(K\setminus G/K\) of double cosets of \(G\) with respect to \(K\). In this paper we show that the group \(B_\infty\), of the braids with finitely many crossings on infinitely many strands, admits such a structure.Around braidshttps://www.zbmath.org/1483.200702022-05-16T20:40:13.078697Z"Vershinin, Vladimir"https://www.zbmath.org/authors/?q=ai:vershinin.vladimir-vBounded Engel elements in residually finite groupshttps://www.zbmath.org/1483.200712022-05-16T20:40:13.078697Z"Bastos, Raimundo"https://www.zbmath.org/authors/?q=ai:bastos.raimundo|bastos.raimundo-a"Silveira, Danilo"https://www.zbmath.org/authors/?q=ai:silveira.daniloSummary: Let \(q\) be a prime. Let \(G\) be a residually finite group satisfying an identity. Suppose that for every \(x \in G\) there exists a \(q\)-power \(m=m(x)\) such that the element \(x^m\) is a bounded Engel element. We prove that \(G\) is locally virtually nilpotent. Further, let \(d, n\) be positive integers and \(w\) a non-commutator word. Assume that \(G\) is a \(d\)-generator residually finite group in which all \(w\)-values are \(n\)-Engel. We show that the verbal subgroup \(w(G)\) has \(\{d,n,w\}\)-bounded nilpotency class.Left \(3\)-Engel elements of odd order in groupshttps://www.zbmath.org/1483.200722022-05-16T20:40:13.078697Z"Jabara, Enrico"https://www.zbmath.org/authors/?q=ai:jabara.enrico"Traustason, Gunnar"https://www.zbmath.org/authors/?q=ai:traustason.gunnarSummary: Let \( G\) be a group and let \( x\in G\) be a left 3-Engel element of odd order. We show that \( x\) is in the locally nilpotent radical of \( G\). We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by \textit {A. I. Kostrikin} and \textit {E. I. Zel'manov} [Tr. Mat. Inst. Steklova 183, 106--111 (1990; Zbl 0729.17006)]. We also give some applications of our main result. In particular, for any given word \( w=w(x_1,\ldots ,x_n)\) in \( n\) variables, we show that if the variety of groups satisfying the law \( w^3=1\) is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law \( (x_{n+1}^3w^3)^3=1\).On conjugacy of abstract root bases of root systems of Coxeter groupshttps://www.zbmath.org/1483.200732022-05-16T20:40:13.078697Z"Dyer, Matthew"https://www.zbmath.org/authors/?q=ai:dyer.matthew-jSummary: We introduce and study a combinatorially defined notion of the root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root systems are extended to abstract root bases, to a larger class of real root systems and, with a short list of (genuine) exceptions, to infinite rank irreducible Coxeter systems.Imaginary cone and reflection subgroups of Coxeter groupshttps://www.zbmath.org/1483.200742022-05-16T20:40:13.078697Z"Dyer, Matthew J."https://www.zbmath.org/authors/?q=ai:dyer.matthew-jLet \(V\) be an \(\mathbb{R}\)-vector space equipped with a symmetric bilinear form \(\langle -,- \rangle\). Suppose \((\Phi, \Pi)\) is a based root system in \(V\) with associated Coxeter system \((W,S)\). Denote
\[
\mathscr{C} = \{v \in V \mid \langle v, \alpha \rangle \ge 0, \forall \alpha \in \Pi\}, \text{ and } \mathscr{K} = (\mathbb{R}_{\ge 0} \Pi) \cap (- \mathscr{C}).
\]
Define the imaginary cone \(\mathscr{Z}\) to be
\[
\mathscr{Z} = \bigcup_{w \in W} w \mathscr{K}.
\]
This extends the notion for Kac-Moody Lie algebras studied in [\textit{V. G. Kac}, Infinite dimensional Lie algebras. Cambridge etc.: Cambridge University Press (1990; Zbl 0716.17022)]. The paper under review provides a survey on the imaginary cone, emphasizing its relationship with reflection subgroups. There are four main results in this paper, which were unknown previously, listed as follows.
Theorem 6.3./Theorem 12.2. Let \(W^\prime\) be a reflection subgroup of \(W\), then \(\mathscr{Z}_{W^\prime} \subseteq \mathscr{Z}\), where \(\mathscr{Z}_{W^\prime}\) is the imaginary cone of \(W^\prime\).
Theorem 7.6. Suppose \(W\) is irreducible, infinite, and of finite rank. Then \(\overline{\mathscr{Z}}\) is the unique non-zero \(W\)-invariant closed pointed cone contained in \(\mathbb{R}_{\ge 0} \Pi\).
Theorem 10.3. (sketched) Suppose \(W\) is of finite rank. (a) The imaginary cone and the Tits cone is a dual pair. (b) The lattice (i.e. poset) formed by faces of \(\mathscr{Z}\) is isomorphic to the lattice formed by facial subgroups without finite components. (c) The face lattice of \(\mathscr{Z}\) is dual to that of the Tits cone. (d) If \(W^\prime\) is a facial subgroup without finite components, then its imaginary cone and its Tits cone can be described explicitly by each other.
Theorem 12.3. One has \(\mathscr{Z} = \mathbb{R}_{\ge 0} (\bigcup_{W^\prime \in \daleth} \mathscr{Z}_{W^\prime})\), where \(\daleth\) is the set of dihedral reflection subgroups of \(W\).
Besides, the hyperbolic and universal cases are discussed in Section 9. In Section 13, some motivations and applications are presented, including the dominance order, limit roots, etc. In particular, the author mentioned that a weakened version of the boundedness conjecture on Lusztig's \(a\)-function can be proved, but no more details are given. There is a list of notations at the end, which is helpful in reading.
Reviewer: Hongsheng Hu (Beijing)Semi-direct decompositions of Coxeter groupshttps://www.zbmath.org/1483.200752022-05-16T20:40:13.078697Z"Guo, Xiangqian"https://www.zbmath.org/authors/?q=ai:guo.xiangqian"Liu, Xuewen"https://www.zbmath.org/authors/?q=ai:liu.xuewen(no abstract)Right-angled Coxeter quandles and polyhedral productshttps://www.zbmath.org/1483.200762022-05-16T20:40:13.078697Z"Kishimoto, Daisuke"https://www.zbmath.org/authors/?q=ai:kishimoto.daisukeSummary: To a Coxeter group \(W\) one associates a quandle \(X_W\) from which one constructs a group \(\mathrm{Ad}(X_W)\). This group turns out to be an intermediate object between \(W\) and the associated Artin group. By using a result of \textit{T. Akita} [Kyoto J. Math. 60, No. 4, 1245--1260 (2020; Zbl 07286664)], we prove that \(\mathrm{Ad}(X_W)\) is given by a pullback involving \(W\), and by using this pullback, we show that the classifying space of \(\mathrm{Ad}(X_W)\) is given by a space called a polyhedral product whenever \(W\) is right-angled. Two applications of this description of the classifying space are given.Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contractionhttps://www.zbmath.org/1483.200772022-05-16T20:40:13.078697Z"Arzhantseva, Goulnara N."https://www.zbmath.org/authors/?q=ai:arzhantseva.goulnara-n"Cashen, Christopher H."https://www.zbmath.org/authors/?q=ai:cashen.christopher-h"Gruber, Dominik"https://www.zbmath.org/authors/?q=ai:gruber.dominik"Hume, David"https://www.zbmath.org/authors/?q=ai:hume.david.2Summary: We introduce and begin a systematic study of sublinearly contracting projections.
We give two characterizations of Morse quasi-geodesics in an arbitrary geodesic metric space. One is that they are sublinearly contracting; the other is that they have completely superlinear divergence.
We give a further characterization of sublinearly contracting projections in terms of projections of geodesic segments.The \(L^2\)-torsion polytope of amenable groupshttps://www.zbmath.org/1483.200782022-05-16T20:40:13.078697Z"Funke, Florian"https://www.zbmath.org/authors/?q=ai:funke.florianSummary: We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the \(L^2\)-torsion polytope among \(G\)-CW-complexes for these groups. As another application we prove that the \(L^2\)-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.Subgroup distortion of 3-manifold groupshttps://www.zbmath.org/1483.200792022-05-16T20:40:13.078697Z"Nguyen, Hoang Thanh"https://www.zbmath.org/authors/?q=ai:nguyen.hoang-thanh"Sun, Hongbin"https://www.zbmath.org/authors/?q=ai:sun.hongbin.1|sun.hongbinIn this paper, the authors characterize the subgroup distortion of all finitely generated subgroups of all finitely-generated 3-manifold groups. In particular, they show that the distortion can only be linear quadratic, exponential, or doubley exponential.
The distortion of a subgroup essentially measures the difference between the word length metric in the subgroup itself and in the ambient group. Understanding this distortion is a basic question in geometric group theory, and the situation is known in many cases for geometric 3-manifolds. This paper first completes an open case to finish the list of all possible geometries when the 3-manifold has empty or tori boundary.
The main theorem of the paper is to characterize the distortion for all finitely generated 3-manifold groups, including nongeometric ones. The method of proof is to extend the notion of a previous paper of the second author of the ``almost fiber surface'' of the subgroup. This is a surface embedded in the covering space of the manifold associated with the subgroup. It turns out that the distortion of the subgroup is related to the separability of the fundamental group of (components of) the almost fiber surface. Thus, by characterizing the almost fiber surface, the paper characterizes the distortion of the subgroup.
This is clearly a powerful technique and an interesting application. Although technical arguments in geometric group can be rather detailed, this paper is well-written and should be of interest to anyone in geometric group theory or low dimensional topology.
Reviewer: Alden Walker (Chicago)Characterization of simple symplectic groups of degree 4 over locally finite fields in the class of periodic groupshttps://www.zbmath.org/1483.200852022-05-16T20:40:13.078697Z"Lytkina, D. V."https://www.zbmath.org/authors/?q=ai:lytkina.daria-v"Mazurov, V. D."https://www.zbmath.org/authors/?q=ai:mazurov.victor-danilovichSummary: Let \(G\) be a periodic group containing an element of order 2 such that each of its finite subgroups of even order lies in a finite subgroup isomorphic to a simple symplectic group of degree 4. It is shown that \(G\) is isomorphic to a simple symplectic group \(S_4(Q)\) of degree 4 over some locally finite field \(Q\).Presentations of affine Kac-Moody groupshttps://www.zbmath.org/1483.200872022-05-16T20:40:13.078697Z"Capdeboscq, Inna"https://www.zbmath.org/authors/?q=ai:capdeboscq.inna"Kirkina, Karina"https://www.zbmath.org/authors/?q=ai:kirkina.karina"Rumynin, Dmitriy"https://www.zbmath.org/authors/?q=ai:rumynin.dmitriy-aSummary: How many generators and relations does \(\mathrm{SL}\,_n(\mathbb{F}_q[t,t^{-1}])\) need? In this paper we exhibit its explicit presentation with 9 generators and 44 relations. We investigate presentations of affine Kac-Moody groups over finite fields. Our goal is to derive finite presentations, independent of the field and with as few generators and relations as we can achieve. It turns out that any simply connected affine Kac-Moody group over a finite field has a presentation with at most 11 generators and 70 relations. We describe these presentations explicitly type by type. As a consequence, we derive explicit presentations of Chevalley groups \(G(\mathbb{F}_q[t,t^{-1}])\) and explicit profinite presentations of profinite Chevalley groups \(G(\mathbb{F}_q[[t]])\).On property (T) for \(\Aut(F_n)\) and \(\mathrm{SL}_n(\mathbb{Z})\)https://www.zbmath.org/1483.220062022-05-16T20:40:13.078697Z"Kaluba, Marek"https://www.zbmath.org/authors/?q=ai:kaluba.marek"Kielak, Dawid"https://www.zbmath.org/authors/?q=ai:kielak.dawid"Nowak, Piotr"https://www.zbmath.org/authors/?q=ai:nowak.piotr-wIn the 1960s, \textit{D. A. Kazhdan} [Funct. Anal. Appl. 1, 63--65 (1967; Zbl 0168.27602); translation from Funkts. Anal. Prilozh. 1, No. 1, 71--74 (1967)] introduced the original definition of Property (T). It was stated in a representation-theoretic way. Kazhdan showed that a locally compact group with property (T) is compactly generated. Moreover, he showed that a lattice \(\Gamma\) in a locally compact group \(G\) has Property (T) if and only if so does \(G\).
Now, it is known that there are several equivalent conditions for groups to have Property (T). So far, Property (T) has actively been studied by a large number of authors, and has made brilliant progress. Today, the study of Property (T) includes a diverse range of research fields in mathematics, for example group theory, representation theory, differential geometry, the theory of group cohomology, geometric group theory, graph theory, ergodic theory and so on. For motivated readers, there is a remarkable detailed textbook by \textit{B. Bekka} et al. [Kazhdan's property. Cambridge: Cambridge University Press (2008; Zbl 1146.22009)].
Let \(F_n\) be the free group of rank \(n\), and \(\Aut F_n\) the automorphism group of \(F_n\). In this landmark paper, the authors showed that \(\Aut F_n\) has Property (T) for \(n \geq 6\).
Historically, the automorphism groups of free groups were begun to study by Dehn and Nielsen in the 1910s
from a viewpoint of the low dimensional topology. In particular, Nielsen gave the first finite presentations for it.
Over the last one century, multiple facets of the automorphism groups of free groups have been studied by a large number of authors, being compared with important groups including the mapping class groups of surfaces, the braid groups, the general linear groups over the integers and so on.
For the special linear groups over the integers, it is well-known that \(\mathrm{SL}(n,\mathbb Z)\) has Property (T) for \(n \geq 3\) due to Kazhdan since \(\mathrm{SL}(n,\mathbb Z)\) is a lattice in \(\mathrm{SL}(n,\mathbb R)\) having Property (T) for \(n \geq 3\). On the other hand, this fact was also shown directly by \textit{Y. Shalom} [Publ. Math., Inst. Hautes Étud. Sci. 90, 145--168 (1999; Zbl 0980.22017)] who gave an explicit Kazhdan constant for \(\mathrm{SL}(n,\mathbb Z)\) by using a notion of bounded generation.
The group \(\Aut F_n\) is often compared with the general linear group \(\mathrm{GL}(n,\mathbb Z)\)
through the natural surjection \(\rho : \Aut F_n \rightarrow \mathrm{GL}(n,\mathbb Z)\) induced from the abelianization of \(F_n\). The group \(\Aut F_2\) does not have Property (T) since \(\Aut F_2\) surjects onto \(\mathrm{GL}(2,\mathbb Z)\) which does not have Property (T).
For \(n=3\), the fact that \(\Aut F_3\) does not have Property (T) is obtained from independent works of
\textit{J. McCool} [Math. Proc. Camb. Philos. Soc. 106, No. 2, 207--213 (1989; Zbl 0733.20031)], and \textit{F. Grunewald} and \textit{A. Lubotzky} [Geom. Funct. Anal. 18, No. 5, 1564--1608 (2009; Zbl 1175.20028)].
For \(n=4\), the problem is still open. \textit{M. Kaluba} et al. [Math. Ann. 375, No. 3--4, 1169--1191 (2019; Zbl 07126529)] showed that \(\Aut F_5\) has Property (T). Combining with these former results and the main result of the paper, we see that \(\Aut F_n\) has Property (T) for \(n\ge 5\).
In this paper, the authors adopt the following definition of Property (T) due to \textit{N. Ozawa} [J. Inst. Math. Jussieu 15, No. 1, 85--90 (2016; Zbl 1336.22008)]. Let \(G\) be a group with a finite symmetric generating set \(S\). In the real group algebra \(\mathbb R[G]\) of \(G\), the element
\[ \Delta := |S|- \sum_{s \in S} s= \frac{1}{2} \sum_{s \in S} (1-s)^*(1-s) \]
is called the Laplacian of \(G\) with respect to \(S\) where the map \(* : \mathbb R[G] \rightarrow \mathbb R[G]\) is induced by \(g \mapsto g^{-1}\) for any \(g \in G\). The group \(G\) is said to have Property (T) if there exist \(\lambda>0\) and finitely many elements \(\xi_i \in\mathbb R[G]\) such that
\[ \Delta^2- \lambda \Delta=\sum_i \xi_i^* \xi_i. \]
Let \(\mathrm{SAut}\,F_n\) be the preimage of \(\mathrm{SL}(n,\mathbb Z)\) by \(\rho\). It is called the special automorphism group of \(F_n\), and is of index \(2\) in \(\Aut F_n\). It has a finite presentation whose generators are all Nielsen transvections due to \textit{S. M. Gersten} [J. Pure Appl. Algebra 33, 269--279 (1984; Zbl 0542.20021)].
In this paper, for \(G=\mathrm{SAut}\,F_n\) and \(S\) being set of all Nielsen transvections, the authors give an explicit estimate on Kazhdan constants and show that the Kazhdan radius is at most \(2\). By using it, the authors prove that \(\mathrm{SAut}\,F_n\) has Property (T) for \(n \geq 6\).
As a corollary, it is seen that \(\Aut F_n\) and the outer automorphism group \(\mathrm{Out}\,F_n\) have Property (T) for \(n \geq 6\).
The authors' technique can be applied to the case where \(G=\mathrm{SL}(n,\mathbb Z)\) and \(S\) is the set of all elementary matrices for \(n \geq 3\). This means that the authors give a new proof for the fact that
\(\mathrm{SL}(n,\mathbb Z)\) has Property (T) for \(n \geq 3\).
This excellent work by the authors will hold a place in the page of history for the study of the automorphism groups of free groups.
Reviewer: Takao Satoh (Tokyo)Computational bounds for doing harmonic analysis on permutation modules of finite groupshttps://www.zbmath.org/1483.430102022-05-16T20:40:13.078697Z"Hansen, Michael"https://www.zbmath.org/authors/?q=ai:hansen.michael-alan|hansen.michael-edberg|hansen.michael-reichhardt|hansen.michael-pilegaard"Koyama, Masanori"https://www.zbmath.org/authors/?q=ai:koyama.masanori"McDermott, Matthew B. A."https://www.zbmath.org/authors/?q=ai:mcdermott.matthew-b-a"Orrison, Michael E."https://www.zbmath.org/authors/?q=ai:orrison.michael-e"Wolff, Sarah"https://www.zbmath.org/authors/?q=ai:wolff.sarahSummary: We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of permutation modules, and it uses the multiplicities of irreducible submodules within individual orbital spaces to express the resulting computational bounds. We conclude by showing that these bounds are surprisingly small when dealing with certain permutation modules arising from the action of the symmetric group on tabloids.Quasicircle boundaries and exotic almost-isometrieshttps://www.zbmath.org/1483.530652022-05-16T20:40:13.078697Z"Lafont, Jean-François"https://www.zbmath.org/authors/?q=ai:lafont.jean-francois"Schmidt, Benjamin"https://www.zbmath.org/authors/?q=ai:schmidt.benjamin"van Limbeek, Wouter"https://www.zbmath.org/authors/?q=ai:van-limbeek.wouterSummary: We show that the limit set of an isometric and convex cocompact action of a surface group on a proper geodesic hyperbolic metric space, when equipped with a visual metric, is a Falconer-Marsh (weak) quasicircle. As a consequence, the Hausdorff dimension of such a limit set determines its bi-Lipschitz class. We give applications, including the existence of almost-isometries between periodic negatively curved metrics on \(\mathbb{H}^2\) that cannot be realized equivariantly.Handlebody bundles and polytopeshttps://www.zbmath.org/1483.570152022-05-16T20:40:13.078697Z"Hensel, Sebastian"https://www.zbmath.org/authors/?q=ai:hensel.sebastian-wolfgang"Kielak, Dawid"https://www.zbmath.org/authors/?q=ai:kielak.dawidThe authors construct an infinite family of non-diffeomorphic three-manifolds with first Betti number \(\geq 2\) and which fiber over the circle such that all the monodromies of the fibered faces of the corresponding Thurston polytope extend from the closed surface on which they are defined to a handlebody. The proof of this result relies on a connection between handlebody bundles and free-by-cyclic groups.
The result establishes a connection between mapping classes of surfaces and outer automorphisms of free groups, namely, to every such automorphism one can associate (infinitely many) mapping classes which inherit properties from the free group automorphism.
Reviewer: Athanase Papadopoulos (Strasbourg)On type-preserving representations of thrice punctured projective plane grouphttps://www.zbmath.org/1483.570232022-05-16T20:40:13.078697Z"Maloni, Sara"https://www.zbmath.org/authors/?q=ai:maloni.sara"Palesi, Frédéric"https://www.zbmath.org/authors/?q=ai:palesi.frederic"Yang, Tian"https://www.zbmath.org/authors/?q=ai:yang.tian|yang.tian.1In this paper, the authors study type-preserving representations of a punctured surface \(S\) into the group of isometries of the hyperbolic space \({\mathbb H}^2\). More precisely, they consider the space \(\mathcal{X}(S)\) of type-preserving representations up to conjugation by \(PSL(2;{\mathbb R}) = SL(2;{\mathbb R})/\{\pm Id\}\). They are interested in questions of \textit{R. M. Kashaev} [Math. Res. Lett. 12, No. 1, 23--36 (2005; Zbl 1082.32011)] on the number of connected components of \(\mathcal{X}(S)\), of \textit{B. H. Bowditch} [Proc. Lond. Math. Soc. (3) 77, No. 3, 697--736 (1998; Zbl 0928.11030)] on the existence of representations such that the image of all \(2\)-sided simple closed curves is hyperbolic but which are not discrete and faithful, and of \textit{W. M. Goldman} [Proc. Symp. Pure Math. 74, 189--214 (2006; Zbl 1304.57025)] on the ergodicity of the action of the mapping class group Mod\((S)\) on \(\mathcal{X}(S)\). They answer these three questions in the case of the thrice-punctured projective plane \(N_{1,3}\).
Reviewer: Leila Ben Abdelghani (Monastir)Fundamental groups of aspherical manifolds and maps of non-zero degreehttps://www.zbmath.org/1483.570242022-05-16T20:40:13.078697Z"Neofytidis, Christoforos"https://www.zbmath.org/authors/?q=ai:neofytidis.christoforosSummary: We define a new class of irreducible groups, called groups not infinite-indexpresentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Löh, providing new classes of aspherical manifolds -- beyond those non-positively curved ones which were predicted by Gromov -- that do not admit maps of non-zero degree from direct products.
A sample application is that an aspherical geometric 4-manifold admits a map of non-zero degree from a direct product if and only if it is a virtual product itself. This completes a characterization of the product geometries due to Hillman. Along the way we prove that for certain groups the property IIPP is a criterion for reducibility. This especially implies the vanishing of the simplicial volume of the corresponding aspherical manifolds. It is shown that aspherical manifolds with reducible fundamental groups do always admit maps of non-zero degree from direct products.\(O_n\) is an \(n\)-MCFLhttps://www.zbmath.org/1483.681692022-05-16T20:40:13.078697Z"Gebhardt, Kilian"https://www.zbmath.org/authors/?q=ai:gebhardt.kilian"Meunier, Frédéric"https://www.zbmath.org/authors/?q=ai:meunier.frederic"Salvati, Sylvain"https://www.zbmath.org/authors/?q=ai:salvati.sylvainSummary: Commutative properties in formal languages pose problems at the frontier of computer science, computational linguistics and computational group theory. A prominent problem of this kind is the position of the language \(O_n\), the language that contains the same number of letters \(a_i\) and \(\overline{a}_i\) with \(1 \leq i \leq n\), in the known classes of formal languages. It has recently been shown that \(O_n\) is a Multiple Context-Free Language (MCFL). However the more precise conjecture of Nederhof that \(O_n\) is an MCFL of dimension \(n\) was left open. We prove this conjecture using tools from algebraic topology. On our way, we prove a variant of the necklace splitting theorem.