Recent zbMATH articles in MSC 22https://www.zbmath.org/atom/cc/222022-05-16T20:40:13.078697ZWerkzeugDynamics of monad graph of finite grouphttps://www.zbmath.org/1483.050742022-05-16T20:40:13.078697Z"Algam, Akram Ghanim"https://www.zbmath.org/authors/?q=ai:algam.akram-ghanim"Shelash, Hayder Baqer"https://www.zbmath.org/authors/?q=ai:shelash.hayder-baqerSummary: Monad graph was introduced by \textit{V. I. Arnold} [Funct. Anal. Appl. 37, No. 3, 177--190 (2003; Zbl 1040.05015); translation from Funkts. Anal. Prilozh. 37, No. 3, 20--35 (2003)], and many result were investigated in . In this paper is we study the properties of monad graph generated by dynamical system defined on finit group which is isomorphic to \(C_n\) cyclic group of order \(n\) and Dihedral group \(D_{2n} \).Eisenstein cohomology and automorphic \(L\)-functionshttps://www.zbmath.org/1483.111022022-05-16T20:40:13.078697Z"Grbac, Neven"https://www.zbmath.org/authors/?q=ai:grbac.nevenThe article under review is a very nice survey of the author's collaborative work with Joachim Schwermer, carried out over the past 12 years on the aspects related to cohomology of arithmetic groups. More precisely, the author took the reader for a quick overview on their work on ``arithmetic groups, Eisenstein cohomology, automorphic forms and their associated \(L\)-functions'', and discussed the main results obtained in the series of articles [the author and \textit{J. Schwermer}, C. R., Math., Acad. Sci. Paris 348, No. 11--12, 597--600 (2010; Zbl 1203.11043); in: Arithmetic geometry and automorphic forms. Festschrift dedicated to Stephen Kudla on the occasion of his 60th birthday. Somerville, MA: International Press; Beijing: Higher Education Press. 209--252 (2011; Zbl 1272.11071); Int. Math. Res. Not. 2011, No. 7, 1654--1705 (2011; Zbl 1298.11050); Forum Math. 26, No. 6, 1635--1662 (2014; Zbl 1317.11055); Forum Math. 31, No. 5, 1225--1263 (2019; Zbl 1439.11133); Adv. Math. 376, Article ID 107438, 49 p. (2021; Zbl 1459.11128)].
The large part of their collaborative work have been focused on studying the automorphic cohomology associated to the symplectic \(\mathrm{Sp}_n\) and unitary \(\mathrm{SU}(n, 1)\) groups and in unraveling the information of geometric and of arithmetic nature. In this survey article, the author mainly discussed their results about the \(\mathbb{Q}\)-rank \(n\) symplectic group \(\mathrm{Sp}_n\) defined over \(\mathbb{Q}\). This can be found in Sections 4 and 5 the article.
This is undoubtedly a very rich area of research within pure mathematics and lies in the interface of representation theory, arithmetic and algebraic geometry, topology (of underlying locally symmetric spaces associated to arithmetic group under consideration), and number theory in a very broad sense.
Eisenstein cohomology and special values of associated \(L\)-functions have numerous applications in various areas of pure mathematics which have been already proven through developments in number theory and especially to Langlands programme over the past 50 years. See Harder's ICM report [\textit{G. Harder}, in: Proceedings of the international congress of mathematicians (ICM), August 21-29, 1990, Kyoto, Japan. Volume II. Tokyo etc.: Springer-Verlag. 779--790 (1991; Zbl 0752.11023)] for more details on the relation of Eisenstein cohomology with other topics. The study of Eisenstein cohomology was initiated by \textit{G. Harder} [Invent. Math. 89, 37--118 (1987; Zbl 0629.10023)], and discovered that this is related to several important topics in number theory, e.g., special values of \(L\)-functions, extension of motives, etc.
For the entire collection see [Zbl 1401.20003].
Reviewer: Jitendra Bajpaj (Göttingen)Square root \(p\)-adic \(L\)-functions. I: Construction of a one-variable measurehttps://www.zbmath.org/1483.111092022-05-16T20:40:13.078697Z"Harris, Michael"https://www.zbmath.org/authors/?q=ai:harris.michael-f|harris.michael-howard|harris.michael-t|harris.michael-p|harris.michael-gSummary: The Ichino-Ikeda conjecture, and its generalization to unitary groups by
\textit{R. N. Harris} [Int. Math. Res. Not. 2014, No. 2, 303--389 (2014; Zbl 1322.11047)], gives explicit formulas for central critical values of a large class of Rankin-Selberg tensor products. The latter conjecture has been proved in full generality and applies to \(L\)-values of the form \(L\big( \frac{1}{2},\mathrm{BC}(\pi)\times\mathrm{BC}(\pi^{\prime})\big)\), where \(\pi\) and \(\pi^{\prime}\) are cohomological automorphic representations of unitary groups \(U(V)\) and \(U(V^{\prime})\), respectively. Here \(V\) and \(V^{\prime}\) are hermitian spaces over a CM field, \(V\) of dimension \(n, V^{\prime}\) of codimension 1 in \(V\), and \(\mathrm{BC}\) denotes the twisted base change to \(\mathrm{GL}(n) \times \mathrm{GL}(n-1)\).
This paper contains the first steps toward constructing a \(p\)-adic interpolation of the normalized square roots of these \(L\)-values, generalizing the construction in my paper with and \textit{J. Tilouine} [Math. Ann. 320, No. 1, 127--147 (2001; Zbl 1034.11034)] on triple product \(L\)-functions. It will be assumed that the CM field is imaginary quadratic, \(\pi\) is a holomorphic representation and \(\pi^{\prime}\) varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to \(\pi\) uses recent work of Eischen, Fintzen, Mantovan, and Varma.Polynomial tau-functions for the multicomponent KP hierarchyhttps://www.zbmath.org/1483.140852022-05-16T20:40:13.078697Z"Kac, Victor G."https://www.zbmath.org/authors/?q=ai:kac.victor-g"De Leur, Johan W. van"https://www.zbmath.org/authors/?q=ai:van-de-leur.johan-wIn a previous paper [Jpn. J. Math. (3) 13, No. 2, 235--271 (2018; Zbl 1401.14209)], the authors constructed all polynomial tau-functions of the \(1\)-component KP hierarchy, namely, they showed that any such tau-function is obtained from a Schur polynomial \(s_\lambda(t)\) by certain shifts of arguments. In the present paper they give a simpler proof of this result, using the (\(1\)-component) boson-fermion correspondence. Moreover, they show that this approach can be applied to the \(s\)-component KP hierarchy, using the \(s\)-component boson-fermion correspondence, finding thereby all its polynomial tau-functions. The authors also find all polynomial tau-functions for the reduction of the \(s\)-component KP hierarchy, associated to any partition consisting of \(s\) positive parts.
The paper is organized as follows. The first section is an introduction to the subject. Section 2 is devoted to the fermionic formulation of the KP hierarchy and Section 3 to the bosonic formulation of KP. Section 4 deals with polynomial solutions of KP. In Section 5 the authors introduce the \(s\)-component KP, where \(s\) is a positive integer. Section 6 is devoted to the \(n\)-KdV where \(n\) is an integer, \(n\geq2\). In Section 7 the authors consider a reduction of the \(s\)-component KP hierarchy, which describes the loop group orbit of \(SL_n\), where \(n=n_1+n_2+\cdots+n_s\), with \(n_1\geq n_2\geq\cdots\geq n_s\geq1\). The case \(s=1\) is the nth Gelfand-Dickey hierarchy. The case \(n=s=2\), i.e., \(n_1=n_2=1\), is the AKNS (or nonlinear Schrödinger) hierarchy. Section 8 is devoted to the AKNS hierarchy.
Reviewer: Ahmed Lesfari (El Jadida)Representations of cohomological Hall algebras and Donaldson-Thomas theory with classical structure groupshttps://www.zbmath.org/1483.141032022-05-16T20:40:13.078697Z"Young, Matthew B."https://www.zbmath.org/authors/?q=ai:young.matthew-bCohomological Hall algebras of quivers with potential were introduced by Kontsevich and Soibelman as tractable algebraic analogues of Donaldson-Thomas invariants of Calabi-Yau threefolds, and as a mathematical model of ``BPS algebras'' studied in theoretical physics by Harvey and Moore. In this context, representations of cohomological Hall algebras are expected to correspond to ``open'' BPS invariants in a purely algebraic setting.
The present paper studies a class of representations of cohomological Hall algebras attached to quivers with involution and equivariant potential. In the geometric incarnation, this corresponds to an extension of Donaldson-Thomas theory from structure group \(GL(n,\mathbb C)\) to the other classical groups. This is expected to model ``orientifold'' BPS invariants. The main result of the paper is an extension of the integrality results of Kontevich-Soibelman and Reineke to this equivariant setting.
The paper fits into the author's larger program on the mathematical underpinnings and repercussions of the orientifold construction.
Reviewer: Johannes Walcher (Montréal)A construction by deformation of unitary irreducible representations of \(\mathrm{SU}(1, n)\) and \(S\mathrm{SU}(n + 1)\)https://www.zbmath.org/1483.170082022-05-16T20:40:13.078697Z"Cahen, Benjamin"https://www.zbmath.org/authors/?q=ai:cahen.benjaminIn the paper under review, the author constructs holomorphic discrete series representations of \(\mathrm{SU}(1, n)\) and some unitary irreducible representations of \(\mathrm{SU}(n)\) by deforming a minimal realization of \(\mathfrak{g}=\mathfrak{sl}(n+ 1,\mathbb C)\). The minimal realization refers to a representation \(\rho_0\) of \(\mathfrak{g}\) in the space of complex polynomials with \(n\) variables, which is given by the classical Weyl correspondence. The term ``minimal'' indicates that the construction of \(\rho_0\) is closely related to the minimal nilpotent coadjoint orbit of \(\mathfrak{g}\). The deformation of \(\rho_0\) is given over the space \(M\) of the complex polynomials with \(2n\) variables and is controlled by the first Chevalley-Eilenberg cohomology space \(H^1(\mathfrak{g},M)\).
Reviewer: Husileng Xiao (Harbin)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.The regular representations of \(\mathrm{GL}_N\) over finite local principal ideal ringshttps://www.zbmath.org/1483.200812022-05-16T20:40:13.078697Z"Stasinski, Alexander"https://www.zbmath.org/authors/?q=ai:stasinski.alexander"Stevens, Shaun"https://www.zbmath.org/authors/?q=ai:stevens.shaunSummary: Let \(\mathfrak{o}\) be the ring of integers in a non-Archimedean local field with finite residue field, \(\mathfrak{p}\) its maximal ideal, and \(r \ge 2\) an integer. An irreducible representation of the finite group \(G_r= \mathrm{GL}_N(\mathfrak{o}/\mathfrak{p}^r)\), for an integer \(N \ge 2\), is called regular if its restriction to the principal congruence kernel \(K^{r-1}=1+\mathfrak{p}^{r-1}M_N(\mathfrak{o}/\mathfrak{p}^r)\) consists of representations whose stabilisers modulo \(K^1\) are centralisers of regular elements in \(M_N(\mathfrak{o}/\mathfrak{p})\).
The regular representations form the largest class of representations of \(G_r\) which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of \(G_r\).Automatic continuity of abstract homomorphisms between locally compact and Polish groupshttps://www.zbmath.org/1483.220012022-05-16T20:40:13.078697Z"Braun, O."https://www.zbmath.org/authors/?q=ai:braun.oliver|braun.oleg-m"Hofmann, Karl H."https://www.zbmath.org/authors/?q=ai:hofmann.karl-heinrich"Kramer, L."https://www.zbmath.org/authors/?q=ai:kramer.linusThe authors introduce the following class \(\mathcal{K}\) of \textit{almost Polish spaces}:
\noindent Every \(X\in\mathcal{K}\) is a Hausdorff topological space, \(\mathcal{K}\) is closed under the passage to closed subspaces and closed under finite products, and the following properties are satisfied: (1) Every open covering of \(X\) has a countable subcovering. (2) The space \(X\) is not a countable union of nowhere dense subsets. (3) For each continuous image \(A\subseteq X\) of some \(\mathcal{K}\)-member there is an open set \(U\subseteq X\) such that the symmetric difference \((A\setminus U)\cap (U\setminus A)\) is a countable union of nowhere dense subsets of \(X\).
The class \(\mathcal{P}\) of Polish spaces, the class \(\mathcal{L}^\sigma\) of locally compact \(\sigma\)-spaces and the class \(\mathcal{C}\) of compact spaces are almost Polish.
If \(\mathcal{K}\) is an almost Polish class and \(X\in\mathcal{K}\), then \(A\subseteq X\) is called a \textit{\(\mathcal{K}\)-analytic set} if \(A=\psi(Z)\) holds for some \(Z\in\mathcal{K}\) and some continuous map \(\psi:Z\to X\).
If \(G\) is a topological group with \(G\in\mathcal{K}\), then \(G\) is called \textit{\(\mathcal{K}\)-rigid} if the following holds:
\noindent For every short exact sequence of groups \[1\to N\hookrightarrow K\overset{\varphi} \to G\to 1,\] where \(\varphi\) is an abstract group homomorphism, where \(G,K\in\mathcal{K}\), and where the kernel \(N\) of \(\varphi\) is \(\mathcal{K}\)-analytic, the homomorphism \(\varphi\) is automatically continuous and open.
The authors show that rigidity fails in the following cases:
\noindent (a) Abelian groups. (b) Groups which are not locally compact or \(\sigma\)-compact. (c) Infinite products of compact Lie groups if the kernel is not restricted. (d) \(\mathrm{SL}(n,\mathbb{C})\) and all infinite complex linear algebraic groups. (e) Certain connected perfect real algebraic groups.
The first main result is Theorem 2.9. Let \(\mathcal{K}\) be a class of almost Polish spaces, and let \(K,G\in\mathcal{K}\) be topological groups. If \(\varphi:K\to G\) is an abstract group homomorphism such that for every identity neighborhood \(U\subseteq G\) there exists an identity neighborhood \(V\subseteq U\) such that \(\varphi^{-1}(V)\) is almost open (which is the case if \(\varphi^{-1}(V)\) is \(\mathcal{K}\)-analytic), then \(\varphi\) is continuous. If \(\varphi\) is in addition surjective, then it is open.
\noindent In this situation, let \(G\) be a Lie group whose Lie algebra is perfect. If \((K,\ker\varphi)\in\mathcal{K}_a\) and if there exists a compact spacious subset \(C\subseteq G\) such that \(\varphi^{-1}(C)\) is \(\mathcal{K}\)-analytic, then \(\varphi\) is continuous and open. (Let \(\mathcal{K}_a\) denote the class of all pairs \((X,A)\), where \(X\in\mathcal{K}\) and \(A\subseteq X\) is \(\mathcal{K}\)-analytic. \(C\) is called \textit{spacious} if some product of finitely many translates of \(CC^{-1}\) has nonempty interior.)
\noindent By setting \(C=G\) this result implies: Let \(G\) be a compact Lie group whose Lie algebra is semisimple. Then \(G\) is rigid within every almost Polish class \(\mathcal{K}\) that contains \(G\).
\noindent This assertion contains results of \textit{R.~R. Kallman} [Adv. Math. 12, 416--417 (1974; Zbl. 0273.22009)] and \textit{P. Gartside} and \textit{B. Pejić} [Topology Appl. 155, 992--999 (2008; Zbl. 1151.54029)].
The second main result is Theorem 4.6: Let \(G\) be a Lie group such that the center of the connected component is finite and the Lie algebra of \(G\) is a direct sum of absolutely simple ideals. If \(\mathcal{K}\) is an almost Polish class containing \(G\), then \(G\) is rigid within \(\mathcal{K}\).
Then the authors consider semiproducts of Lie groups. In Theorem 5.6 a list of such groups is given which are rigid in every Polish class \(\mathcal{K}\) containing them.
In Section 6 the rigidity of topologically finitely generated profinite groups is studied. The main result is Theorem 6.3: Let \(G\) be a topologically finitely generated profinite group, and let \(\mathcal{K}\) be an almost Polish class. If \(G\) is contained in \(\mathcal{K}\), then \(G\) is rigid within \(\mathcal{K}\).
\noindent This Theorem generalizes a result of Gartside and Pejić [loc. cit.].
Theorem 6.3 remains valid if \(G\) is a compact quasi-semisimple group (see Theorem 7.7). (A nontrivial compact group \(S\) with center \(C(S)\) is called \textit{quasi-semisimple} if the commutator subgroup is dense in \(S\) and if \(S/C(S)\) is topologically simple.)
The very interesting paper ends with Theorem 8.1: Let \(G\) be a Lie group whose Lie algebra is perfect and let \(H\) be a topological group. Let \(\psi:G\to H\) be an abstract homomorphism such that there exists a compact spacious set \(C\subseteq G\) with compact \(\overline{\psi(C)}\), then \(\psi\) is continuous.
Reviewer: Dieter Remus (Hagen)Metrical universality for groups (erratum)https://www.zbmath.org/1483.220022022-05-16T20:40:13.078697Z"Doucha, Michal"https://www.zbmath.org/authors/?q=ai:doucha.michalSummary: The aim of this note is to correct the proof of Proposition 2.15 in the original article [Forum Math. 29, No. 4, 847--872 (2017; Zbl 1375.22001)].Tukey order and diversity of free abelian topological groupshttps://www.zbmath.org/1483.220032022-05-16T20:40:13.078697Z"Gartside, Paul"https://www.zbmath.org/authors/?q=ai:gartside.paul-mSummary: For a Tychonoff space \(X\) the \textit{free abelian topological group} over \(X\), denoted \(A(X)\), is the free abelian group on the set \(X\) with the coarsest topology so that for any continuous map of \(X\) into an abelian topological group its canonical extension to a homomorphism on \(A(X)\) is continuous.
We show there is a family \(\mathcal{A}\) of maximal size, \(2^{\mathfrak{c}}\), consisting of separable metrizable spaces, such that if \(M\) and \(N\) are distinct members of \(\mathcal{A}\) then \(A(M)\) and \(A(N)\) are not topologically isomorphic (moreover, \(A(M)\) neither embeds topologically in \(A(N)\) nor is an open image of \(A(N))\). We show there is a chain \(\mathcal{C}=\{M_\alpha:\alpha<\mathfrak{c}^+\}\), of maximal size, of separable metrizable spaces such that if \(\beta < \alpha\) then \(A( M_\beta)\) embeds as a closed subgroup of \(A( M_\alpha)\) but no subspace of \(A( M_\beta)\) is homeomorphic to \(A( M_\alpha)\).
We show that the character (minimal size of a local base at 0) of \(A(M)\) is \(\mathfrak{d}\) (minimal size of a cofinal set in \(\mathbb{N}^{\mathbb{N}})\) for every non-discrete, analytic \(M\), but consistently there is a co-analytic \(M\) such that the character of \(A(M)\) is strictly above \(\mathfrak{d}\).
The main tool used for these results is the Tukey order on the neighborhood filter at 0 in an \(A(X)\), and a connection with the family of compact subsets of an auxiliary space.A classification of the abelian minimal closed normal subgroups of locally compact second-countable groupshttps://www.zbmath.org/1483.220042022-05-16T20:40:13.078697Z"Reid, Colin D."https://www.zbmath.org/authors/?q=ai:reid.colin-dThe author gives a complete list the abelian topologically characteristically simple locally compact second-countable (l.c.s.c.) groups. Each group on this list occur as the monolith of some soluble l.c.s.c. group of derived length at most \(3\). (The monolith of a locally compact group \(G\) is the intersection of all nontrivial closed normal subgroups of \(G\).)
At the same time, the group \(\mathbb{Q}^n\) and its dual \(\widehat{\mathbb{Q}}^n\) (\(n\in\mathbb{N}\)) from this list cannot occur as minimal closed normal subgroups in any \textit{compactly generated} l.c.s.c. group. With the exception of the groups \(\mathbb{Q}^n\) and \(\widehat{\mathbb{Q}}^n\), every abelian topologically characteristically simple l.c.s.c. group occurs as the monolith of a \textit{compactly generated} soluble l.c.s.c. group of derived length at most \(3\).
Let \(p\) be a prime and let \(\mathbb{Q}_p(\kappa)\) be the group of functions from a set of size \(\kappa\) to \(\mathbb{Q}_p\) with all but finitely many values in \(\mathbb{Z}_p\) (it is clear that if \(\kappa\) is finite, then \(\mathbb{Q}_p(\kappa)=\mathbb{Q}_p^\kappa\)). The author suggests the following characterization of the groups \(\mathbb{Q}_p(\kappa)\) (Proposition 3.10): Let \(A\) be an abelian l.c.s.c. group, and let \(p\) be a prime. Then the following are equivalent: (i) \(A\cong\mathbb{Q}_p(\kappa)\) for some \(\kappa\in\mathbb{N}\cup\{\aleph_0\}\); (ii) \(A\) is totally disconnected and torsion-free, and both \(P_p(A)\) and \(pA\) are dense in \(A\); (iii) \(A\) is totally disconnected and torsion-free, \(P_p(A)=A\), and \(\operatorname{div}(A)\) is dense in \(A\). (Here \(P_p(A)\) is the set of all topological \(p\)-elements of \(A\) and \(\operatorname{div}(A)\) is the largest divisible subgroup of \(A\).)
Reviewer: Mikhail Kabenyuk (Kemerovo)On a noncontractible family of representations of the canonical commutation relationshttps://www.zbmath.org/1483.220052022-05-16T20:40:13.078697Z"Kim, Hyunmoon"https://www.zbmath.org/authors/?q=ai:kim.hyunmoonIn this paper, the authors construct an explicit family of irreducible representations of the canonical commutation relations parametrised by the space of pairs of transverse Lagrangian subspaces in the complexification of a symplectic vector space. In particular, a non-contractible family of representations, constructed by Lion-Vergne and parametrised by the space of pairs of transverse real Lagrangian subspaces, is extended to a family parametrised by space of pairs of transverse complex Lagrangian subspaces. For some representations, the operators can be exponentiated to obtain unitary representations of the canonical and Weyl commutation relations, and this allows the reconstruction of previously studied contractible families of representations.
Reviewer: Carlos André (Lisboa)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)On the growth of \(L^2\)-invariants of locally symmetric spaces. II: Exotic invariant random subgroups in rank onehttps://www.zbmath.org/1483.220072022-05-16T20:40:13.078697Z"Abert, Miklos"https://www.zbmath.org/authors/?q=ai:abert.miklos"Bergeron, Nicolas"https://www.zbmath.org/authors/?q=ai:bergeron.nicolas"Biringer, Ian"https://www.zbmath.org/authors/?q=ai:biringer.ian"Gelander, Tsachik"https://www.zbmath.org/authors/?q=ai:gelander.tsachik"Nikolov, Nikolay"https://www.zbmath.org/authors/?q=ai:nikolov.nikolay"Raimbault, Jean"https://www.zbmath.org/authors/?q=ai:raimbault.jean"Samet, Iddo"https://www.zbmath.org/authors/?q=ai:samet.iddoThis paper is a sequel to [Ann. Math. (2) 185, No. 3, 711--790 (2017; Zbl 1379.22006)] by the same authors. The former paper had extended the validity of the Lück approximation theorem to the setting of Benjamini-Schramm convergence for uniformly discrete sequences of lattices \(\Gamma_n\) in a higher-rank symmetric space of noncompact type \(X=G/K\). That is, it proved \(\lim_{n\to\infty}\frac{b_k(\Gamma_n\backslash X)}{\mathrm{Vol}(\Gamma_n\backslash X)}=\beta_k^{(2)}(X)\) under the assumption that \(\Gamma_n\backslash X\) BS-converges to \(X\).
Their approach towards this theorem had been to introduce the notion of invariant random subgroups (IRSs), that is, conjugation-invariant Borel probability measures on \(\mathrm{Sub}(G)\). (\(\mathrm{Sub}(G)\) is the space of closed subgroups with the Chabauty topology.) For a lattice \(\Gamma\subset G\) one has a map \(\Gamma\backslash G\to \mathrm{Sub}(G)\) sending \(\Gamma g\) to \(g^{-1} \Gamma g\), and one can use the finite measure on \(\Gamma\backslash G\) to define an IRS on \(\mathrm{Sub}(G)\). BS-convergence of \(\Gamma_n\backslash X\) to \(X\) is equivalent to convergence of \(\mu_{\Gamma_n}\) to \(\mu_{\mathrm{id}}\) for the weak-*-topology on \(\mathrm{IRS}(G)\). (The latter is compact, so sequences converge up to extraction.)
By ergodic decomposition it suffices to study ergodic IRSs. If \(\mathrm{rank}_{\mathbb R}(G)\ge 2\), then the Nevo-Stuck-Zimmer theorem implies that the only ergodic IRSs are \(\mu_G,\mu_{\mathrm{id}}\) and \(\mu_\Gamma\) for some lattice \(\Gamma\). Moreover, for every sequence of pairwise non-conjugate lattices, \(\mu_{\Gamma_n}\) converges to \(\mu_{\mathrm{id}}\). This was a main ingredient in the proof by the authors of the improved Lück approximation theorem.
If \(\mathrm{rank}_{\mathbb R}(G)=1\), there are much more possibilities for IRSs. First, in this case for a lattice \(\Gamma\subset G\) the Margulis normal subgroup theorem does not apply. There are many normal subgroups of infinite index which yield an IRS. Next, lattices \(\Gamma\) may have epimorphisms to the free group \(F_2\), and by the work of \textit{L. Bowen} [Groups Geom. Dyn. 9, No. 3, 891--916 (2015; Zbl 1358.37011)] there are many exotic IRSs on free groups. Using the epimorphism one obtains then exotic IRSs supported on \(\Gamma\).
The paper under review is devoted to the construction of other uncountable families of IRSs in \(\mathrm{SO}(n,1)=\mathrm{Isom}^+({\mathbb H}^n)\). It follows from the Borel density theorem, that an ergodic IRS \(\mu\not=\mu_G\) is almost-surely discrete. Thus it can be seen as a probability measure on the set of discrete subgroups or equivalently on the set of (framed) hyperbolic manifolds. The authors describe several constructions of random hyperbolic manifolds, which frequently can not be induced by lattices.
One such construction takes two hyperbolic \(n\)-manifolds \(N_0\) and \(N_1\), whose totally geodesic boundaries consist both of the same two copies of some hyperbolic \((n-1)\)-manifold. To each \(\alpha\in\left\{0,1\right\}^{\mathbb Z}\) one obtains a hyperbolic \(n\)-manifold \(N_\alpha\) by glueing copies of \(N_0\) and \(N_1\) according to the pattern prescribed by \(\alpha\). Each shift-invariant measure on \(\left\{0,1\right\}^{\mathbb Z}\) yields a random hyperbolic \(n\)-manifold. This IRS is not induced by a lattice if \(N_0\) and \(N_1\) are not embedded in non-commensurable compact arithmetic \(n\)-manifolds and \(\alpha\) is not supported on a shift-periodic orbit.
Another construction takes a topological surface \(S\) glued from infinitely many pairs of pants along the pattern of an infinite \(3\)-regular tree. Hyperbolic metrics on \(S\) are described by Fenchel-Nielsen coordinates. Choosing Fenchel-Nielsen coordinates randomly from \(\left(0,\infty\right)\times S^1\) one obtains a random hyperbolic surface. For appropriately measures on \(\left(0,\infty\right)\) and the Lebesgue measure on \(S^1\) one obtains IRSs not induced by a lattice.
A further construction takes a subgroup of the mapping class group \(\mathrm{Mod}(\Sigma)\) freely generated by pseudo-Anosov \(\phi_1,\ldots,\phi_n\), such that orbits on Teichmüller space are quasi-convex. For a sequence of words with \(\vert w_i\vert\to\infty\) let \(\Gamma_i\backslash{\mathbb H}^3\) be hyperbolic \(3\)-manifold fibering over \(S^1\) with monodromy \(w_i\). The sequence \(\mu_{\Gamma_i}\) converges (up to extraction) to an IRS \(\mu\). If the words \(w_i\) are chosen appropriately, then \(\mu\) is not induced by a lattice.
All these constructions yield weak-*-limits of sequences \(\mu_{\Gamma_n}\) for lattices \(\Gamma_n\) and the authors ask whether this must be the case for every ergodic IRS \(\mu\not=\mu_G\).
Reviewer: Thilo Kuessner (Augsburg)Corrigendum to: ``Effective equidistribution of translates of large submanifolds in semisimple homogeneous spaces''https://www.zbmath.org/1483.220082022-05-16T20:40:13.078697Z"Ubis, Adrián"https://www.zbmath.org/authors/?q=ai:ubis.adrianFrom the text: We wish to issue a corrigendum for our paper [ibid. 2017, No. 18, 5629--5666 (2017; Zbl 1405.22014)]. As the paper is currently written, there is an error in the proof of Proposition 3.5. Precisely, we assume that \(f^v_* (g\Gamma) = \tilde{f}_*(u_v g\Gamma)\overline{\tilde{f}_*}(u_v g\Gamma)\) has vanishing integral, while the truth is that this integral is just small due to mixing (but not necessarily zero). This error will affect the definition of mixing function (Definition 3.4), and in turn the proofs and statements of Propositions 6.2 and 9.4, which will change the constants in the statements of Theorems 1.3 and 3.6.Tempered homogeneous spaces. IIIhttps://www.zbmath.org/1483.220092022-05-16T20:40:13.078697Z"Benoist, Yves"https://www.zbmath.org/authors/?q=ai:benoist.yves"Kobayashi, Toshiyuki"https://www.zbmath.org/authors/?q=ai:kobayashi.toshiyukiLet \(G\) be a real semisimple algebraic group. Let \(H\) be a real reductive algebraic subgroup of \(G\). The paper under review characterizes the pairs \((G, H)\) such that the natural unitary representation of \(G\) on the Hilbert space \(L^2(G/H)\) is tempered. Using the earlier temperedness criterion given in Theorem 4.1 of the authors in [J. Eur. Math. Soc. (JEMS) 17, No. 12, 3015--3036 (2015; Zbl 1332.22015)], the current paper gives a necessary condition as well as a sufficient condition for the representation \(L^2(G/H)\) to be tempered. See Theorem 1.1. Moreover, the paper also completely lists the pairs \((G, H)\) such that \(L^2(G/H)\) is \textit{not} tempered.
For Part IV, see \url{arxiv:2009.10391}
Reviewer: Chao-Ping Dong (Changsha)Representations induced from cuspidal and ladder representations of classical \(p\)-adic groupshttps://www.zbmath.org/1483.220102022-05-16T20:40:13.078697Z"Bošnjak, Barbara"https://www.zbmath.org/authors/?q=ai:bosnjak.barbaraIn this paper, the author investigates certain parabolically induced (complex) representations of \(p\)-adic symplectic and odd orthogonal groups. She is concerned with the representations parabolically induced from the irreducible representations of the maximal Levi subgroups which are the tensor product of a ladder representation on the general linear part and a cuspidal representation on the classical part of the Levi subgroup. Additionally, there is an assumption that all the exponents in the cuspidal support of the ladder representations are all bigger or equal to 1/2. The criteria for reducibility of the induced representations of this kind are actually already known by the previous work of Lapid and Tadić, but they haven't described the whole composition series of these representations, which is exactly what the author achieves in this paper.
The significance of these results lies in the fact that one may find some interesting unitarizable representations as a subquotients of these representations; moreover, one hopes that these kinds of induced representations would have the same role in the unitary dual for the classical groups as the Speh representations had in the general linear group case.
Reviewer: Marcela Hanzer (Zagreb)Negative definite functions on the infinite dimensional special linear grouphttps://www.zbmath.org/1483.220112022-05-16T20:40:13.078697Z"Rabaoui, Marouane"https://www.zbmath.org/authors/?q=ai:rabaoui.marouaneIn this paper the author investigate the connection between the boundedness of negative definite functions and Kazhdan's property (T) in the framework of Olshanski spherical pairs.
Let \(\mathbb{F}=\mathbb{R}, \mathbb{C}\) or \(\mathbb{H}\) be the quaternion field, and let \(SL_n(\mathbb{F})\) be the special linear group with \(n\geq 3\). Let \(K_n\) be a compact subgroup of \(SL_n(\mathbb{F})\) such that \((SL_n(\mathbb{F}), K_n)\) is a Gelfand pair. Consider the inductive limit as \(n \rightarrow \infty\), the Olshanski spherical pair \((SL_\infty(\mathbb{F}), K_\infty)\). The author proves that the group \(SL_\infty(\mathbb{F})\) has the Kazhdan property (T), from which he deduces that every continuous negative definite function on \(SL_\infty(\mathbb{F})\) is bounded.
An integral representation of \(K_\infty\)-bi-invariant continuous negative definite function on \(SL_\infty(\mathbb{F})\) is also given.
Reviewer: Abdelhamid Boussejra (Kénitra)On the discreteness of states accessible via right-angled paths in hyperbolic spacehttps://www.zbmath.org/1483.300832022-05-16T20:40:13.078697Z"Lessa, Pablo"https://www.zbmath.org/authors/?q=ai:lessa.pablo"Garcia, Ernesto"https://www.zbmath.org/authors/?q=ai:garcia.ernestoSummary: We consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance \(r>0\) along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of \(r>0\) for which the set of orthonormal frames accessible using these transformations is discrete.
In the hyperbolic plane this is equivalent to solving the discreteness problem (see [\textit{J.Gilman}, Geom. Dedicata 201, 139--154 (2019; Zbl 1421.30056)] and the references therein) for a particular one parameter family of two-generator subgroups of \(\mathrm{PSL}_2(\mathbb{R})\). In the three dimensional case we solve this problem for a particular one parameter family of subgroups of the isometry group which have four generators.Global properties of vector fields on compact Lie groups in Komatsu classeshttps://www.zbmath.org/1483.353102022-05-16T20:40:13.078697Z"Kirilov, Alexandre"https://www.zbmath.org/authors/?q=ai:kirilov.alexandre"de Moraes, Wagner A. A."https://www.zbmath.org/authors/?q=ai:de-moraes.wagner-augusto-almeida"Ruzhansky, Michael"https://www.zbmath.org/authors/?q=ai:ruzhansky.michael-vSummary: In this paper, we characterize completely the global hypoellipticity and global solvability in the sense of Komatsu (of Roumieu and Beurling types) of constant-coefficient vector fields on compact Lie groups. We also analyze the influence of perturbations by lower-order terms in the preservation of these properties.Relative entropy and the Pinsker product formula for sofic groupshttps://www.zbmath.org/1483.370052022-05-16T20:40:13.078697Z"Hayes, Ben"https://www.zbmath.org/authors/?q=ai:hayes.benFrom the abstract: ``We continue our study of the outer Pinsker factor for probability measure-preserving actions of sofic groups. Using the notion of local and doubly empirical convergence developed by Austin we prove that in many cases the outer Pinsker factor of a product action is the product of the outer Pinsker factors. Our results are parallel to those of Seward for Rokhlin entropy. We use these Pinsker product formulas to show that if \(X\) is a compact group, and \(G\) is a sofic group with \(G\curvearrowright X\) by automorphisms, then the outer Pinsker factor of \(G\curvearrowright (X,m_X)\) is given as a quotient by a \(G\)-invariant, closed, normal subgroup of \(X\). We use our results to show that if \(G\) is sofic and \(f\in M_n (\mathbb{Z}(G))\) is invertible as a convolution operator \(\ell^2 (G)^{\oplus n}\to \ell^2 (G)^{\oplus n}\), then the action of \(G\) on the Pontryagin dual of \(\mathbb{Z} (G)^{\oplus n}/\mathbb{Z} (G)^{\oplus n}f\) has completely positive measure-theoretic entropy with respect to the Haar measure.''
The definition of sofic group is given and other auxiliary notions are recalled. The relative entropy for actions of sofic groups is defined and its main properties are discussed. The relative outer Pinsker algebra is defined. Some preliminaries on local and doubly empirical convergence, the definition of strong soficity, and applications to actions on compact groups by automorphisms, are given. Actions which are strongly sofic with respect to any sofic approximation of a countable, discrete, and sofic group, are considered with examples. Certain permanence properties of strong soficity are proven. The author also gives a product formula for outer Pinsker factors of strongly sofic actions and uses it in the study of algebraic actions. It is shown that ``the given definition of (upper) relative entropy for actions of sofic groups agrees with the usual definition when the group is amenable''.
Reviewer: Symon Serbenyuk (Kyïv)Spread out random walks on homogeneous spaceshttps://www.zbmath.org/1483.370072022-05-16T20:40:13.078697Z"Prohaska, Roland"https://www.zbmath.org/authors/?q=ai:prohaska.rolandThe setting considered here is that of a homogeneous space \(X=G/\Gamma\) for a \(\sigma\)-compact locally compact metrizable group \(G\) and a discrete subgroup \(\Gamma<G\), with a Borel probability measure \(\mu\) on \(G\) used to define a random walk on \(X\). That is, each step of the random walk chooses an element \(g\in G\) according to \(\mu\) and then moves \(x\in X\) to \(gx\in X\). Here the Markov chain theory is used to carry out a careful analysis under the assumption that the increment function is spread out. In the lattice (finite volume) case a complete picture of the asymptotics of the \(n\)-step distribution is found, and they are shown to equidistribute to Haar measure. Situations in which this equidistribution is exponentially fast or locally uniform relative to the initial point are studied. In the case of infinite volume the recurrence is shown and it is proved a ratio limit theorem for symmetric spread out random walks on homogeneous spaces under a growth condition.
Reviewer: Thomas B. Ward (Newcastle)Nilpotent Cantor actionshttps://www.zbmath.org/1483.370162022-05-16T20:40:13.078697Z"Hurder, Steven"https://www.zbmath.org/authors/?q=ai:hurder.steven-e"Lukina, Olga"https://www.zbmath.org/authors/?q=ai:lukina.olgaA nilpotent Cantor action is a minimal equicontinuous action of a finitely generated group \(\Gamma\) on a Cantor space \(X\), where \(\Gamma\) contains a finitely generated nilpotent subgroup \(\Gamma_0\) of finite index. Nilpotent Cantor actions arise in the classification of renormalizable groups; that is, finitely generated groups which admit a proper self-embedding with image of finite index. These groups arise in the study of laminations with the shape of a compact manifold, and in the classification of generalized Hirsch foliations.
The authors prove that any effective or faithful action of a finitely generated group on a Cantor space, which is continuously orbit equivalent to a nilpotent Cantor action, must itself be a nilpotent Cantor action. As an application, they obtain new invariants of nilpotent Cantor actions under continuous orbit equivalence. The virtual nilpotency class \(vc(\Gamma)\) of a finitely generated virtually nilpotent group \(\Gamma\) is defined as the length of a central series for a torsion-free nilpotent subgroup of finite index. The second main result is that two finitely generated groups admitting continuously orbit equivalent effective Cantor actions have the same virtual nilpotency class.
Reviewer: Mahender Singh (Sahibzada Ajit Singh Nagar)Dynamical zeta functions of Reidemeister typehttps://www.zbmath.org/1483.370342022-05-16T20:40:13.078697Z"Fel'shtyn, Alexander"https://www.zbmath.org/authors/?q=ai:felshtyn.alexander"Ziętek, Malwina"https://www.zbmath.org/authors/?q=ai:zietek.malwinaSummary: In this paper we study dynamical representation theory zeta functions counting numbers of fixed irreducible representations for iterations of group endomorphism. The rationality and functional equation for these zeta functions are proven for several classes of groups. We prove Pólya-Carlson dichotomy between rationality and a natural boundary for analytic behavior of the Reidemeister zeta functions for a large class of automorphisms of infinitely generated abelian groups. We also establish the connection between the Reidemeister zeta function and dynamical representation theory zeta functions under restriction of endomorphism to a subgroup.A bijection of invariant means on an amenable group with those on a lattice subgrouphttps://www.zbmath.org/1483.430052022-05-16T20:40:13.078697Z"Hopfensperger, John"https://www.zbmath.org/authors/?q=ai:hopfensperger.johnLet \(G=\mathbb{R}^{d}\) and \(\Gamma=\mathbb{Z}^{d}\). Grosvenor showed that there exists a natural affine injection \(i\) from \(LIM(\Gamma)\) into \(TLIM(G)\) such that is also surjection, where \(LIM\) denotes for the set of left invariant means and \(TLIM\) denotes the set of all topologically left invariant means. In the present paper the author improves this result for amenable groups $G$ and shows that \(i\) is a surjection if and only if \(\frac{G}{\Gamma}\) is compact.
Reviewer: Amir Sahami (Tehran)Statistics on Lie groups: a need to go beyond the pseudo-Riemannian frameworkhttps://www.zbmath.org/1483.530262022-05-16T20:40:13.078697Z"Miolane, Nina"https://www.zbmath.org/authors/?q=ai:miolane.nina"Pennec, Xavier"https://www.zbmath.org/authors/?q=ai:pennec.xavierSummary: Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group \(G\) is a manifold that carries an additional group structure. Statistics on \textit{Riemannian} manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall by others. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is \textit{compatible with the group structure}, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group \(G\). The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.
For the entire collection see [Zbl 1470.00021].The anomaly flow on nilmanifoldshttps://www.zbmath.org/1483.531162022-05-16T20:40:13.078697Z"Pujia, Mattia"https://www.zbmath.org/authors/?q=ai:pujia.mattia"Ugarte, Luis"https://www.zbmath.org/authors/?q=ai:ugarte.luisIn the present article, the authors study the so-called anomaly flow on 2-step nilmanifolds with respect to any Hermitian connection in the Gauduchon line. This is a coupled flow of a pair of Hermitian metrics defined on a 3-dimensional complex manifold equipped with a (3,0) form and on the fibers of a holomorphic vector bundle over the mentioned complex manifold, respectively.
In particular, they characterize the solutions for a simplified version of the anomaly flow under the hypothesis that the base space is a 2-step nilmanifold of real dimension 6 with the first Betti number greater or equal than 4. Moreover, they study its convergence and they prove that two properties are preserved under the flow. Namely, the balanced condition and the diagonal character of the metrics.
Reviewer: Alberto Rodríguez-Vázquez (A Coruña)Minimum topological group topologieshttps://www.zbmath.org/1483.540222022-05-16T20:40:13.078697Z"Chang, Xiao"https://www.zbmath.org/authors/?q=ai:chang.xiao"Gartside, Paul"https://www.zbmath.org/authors/?q=ai:gartside.paul-mSummary: A Hausdorff topological group topology on a group \(G\) is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on \(G\). For every compact metrizable space \(X\) containing an open \(n\)-cell, \(n\geq 2\), the homeomorphism group \(H(X)\) has no minimum group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space \(X\) containing a dense open one-manifold, \(H(X)\) has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology.Arithmeticity of hyperbolic \(3\)-manifolds containing infinitely many totally geodesic surfaceshttps://www.zbmath.org/1483.570172022-05-16T20:40:13.078697Z"Mohammadi, Amir"https://www.zbmath.org/authors/?q=ai:mohammadi.amir"Margulis, Gregorii"https://www.zbmath.org/authors/?q=ai:margulis.gregory-aThe main result in this paper is that if a closed hyperbolic 3-manifold \(M\) contains infinitely many totally geodesic surfaces, then \(M\) is arithmetic. The result answers affirmatively an open question asked by Reid and by McMullen, cf. [\textit{D. B. McReynolds} and \textit{A. W. Reid}, Math. Res. Lett. 21, No. 1, 169--185 (2014; Zbl 1301.53039) and \textit{K. Delp} et al., ``Problems In Groups, Geometry, and Three-Manifolds'', Preprint, \url{arXiv:1512.04620}]. The proof of arithmeticity uses a superrigidity theorem. As a consequence, the authors obtain that if \(M = \mathbb{H}^3/\Gamma\) is a closed hyperbolic 3-manifold which contains infinitely many totally geodesic surfaces, the index of \(\Gamma\) in its commensurator group is infinite.
Reviewer: Athanase Papadopoulos (Strasbourg)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.An abstract theory of physical measurementshttps://www.zbmath.org/1483.810152022-05-16T20:40:13.078697Z"Resende, Pedro"https://www.zbmath.org/authors/?q=ai:resende.pedroSummary: The question of what should be meant by a measurement is tackled from a mathematical perspective whose physical interpretation is that a measurement is a fundamental process via which a finite amount of classical information is produced. This translates into an algebraic and topological definition of \textit{measurement space} that caters for the distinction between quantum and classical measurements and allows a notion of observer to be derived.A geometrical representation of the quantum information metric in the gauge/gravity correspondencehttps://www.zbmath.org/1483.810362022-05-16T20:40:13.078697Z"Tsuchiya, Asato"https://www.zbmath.org/authors/?q=ai:tsuchiya.asato"Yamashiro, Kazushi"https://www.zbmath.org/authors/?q=ai:yamashiro.kazushiSummary: We study a geometrical representation of the quantum information metric in the gauge/gravity correspondence. We consider the quantum information metric that measures the distance between the ground states of two theories on the field theory side, one of which is obtained by perturbing the other. We show that the information metric is represented by a back reaction to the volume of a codimension-2 surface on the gravity side if the unperturbed field theory possesses the Poincare symmetry.Symmetry enriched phases of quantum circuitshttps://www.zbmath.org/1483.810392022-05-16T20:40:13.078697Z"Bao, Yimu"https://www.zbmath.org/authors/?q=ai:bao.yimu"Choi, Soonwon"https://www.zbmath.org/authors/?q=ai:choi.soonwon"Altman, Ehud"https://www.zbmath.org/authors/?q=ai:altman.ehudSummary: Quantum circuits consisting of random unitary gates and subject to local measurements have been shown to undergo a phase transition, tuned by the rate of measurement, from a state with volume-law entanglement to an area-law state. From a broader perspective, these circuits generate a novel ensemble of quantum many-body states at their output. In this paper, we characterize this ensemble and classify the phases that can be established as steady states. Symmetry plays a nonstandard role in that the physical symmetry imposed on the circuit elements does not on its own dictate the possible phases. Instead, it is extended by dynamical symmetries associated with this ensemble to form an enlarged symmetry.
Thus, we predict phases that have no equilibrium counterpart and could not have been supported by the physical circuit symmetry alone. We give the following examples. First, we classify the phases of a circuit operating on qubit chains with \(\mathbb{Z}_2\) symmetry. One striking prediction, corroborated with numerical simulation, is the existence of distinct volume-law phases in one dimension, which nonetheless support true long-range order. We furthermore argue that owing to the enlarged symmetry, this system can in principle support a topological area-law phase, protected by the combination of the circuit symmetry and a dynamical permutation symmetry. Second, we consider a Gaussian fermionic circuit that only conserves fermion parity. Here the enlarged symmetry gives rise to a \(U(1)\) critical phase at moderate measurement rates and a Kosterlitz-Thouless transition to area-law phases. We comment on the interpretation of the different phases in terms of the capacity to encode quantum information. We discuss close analogies to the theory of spin glasses pioneered by Edwards and Anderson as well as crucial differences that stem from the quantum nature of the circuit ensemble.Autocorrelation functions for quantum particles in supersymmetric Pöschl-Teller potentialshttps://www.zbmath.org/1483.810802022-05-16T20:40:13.078697Z"Cellarosi, Francesco"https://www.zbmath.org/authors/?q=ai:cellarosi.francescoSummary: We consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric Pöschl-Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fernández provides large classes of Pöschl-Teller partner potentials to which our analysis applies.The ladder operators on \(1+1\)-de Sitter spacehttps://www.zbmath.org/1483.810892022-05-16T20:40:13.078697Z"Rabeie, A."https://www.zbmath.org/authors/?q=ai:rabeie.a"Rezaei, S."https://www.zbmath.org/authors/?q=ai:rezaei.sadegh|rezaei.seyed-saeed-changi|rezaei.sara|rezaei.shahed|rezaei.shahram|rezaei.shadi|rezaei.shahrbanoo|rezaei.seyed-mehdi|rezaei.shayesteh|rezaei.saeidSummary: In this paper, we present the annihilation and creation operators for a moving scalar massive particle on \(1+1\)-de Sitter space. This presentation is based on coherent states method and Hall-Mitchell approach about annihilation operator for a system which its phase space is \(S_{\mathcal{C}}^n \). We show that these operators coincide with the Ladder operators for a quantum particle on circle which was presented by Kowalski-Rembielinski-Papaloucas (both cases have the same phase spaces).Spinorial Snyder and Yang models from superalgebras and noncommutative quantum superspaceshttps://www.zbmath.org/1483.810912022-05-16T20:40:13.078697Z"Lukierski, Jerzy"https://www.zbmath.org/authors/?q=ai:lukierski.jerzy"Woronowicz, Mariusz"https://www.zbmath.org/authors/?q=ai:woronowicz.mariuszSummary: The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic phase superspaces (SUSY Yang model). In last Section we recall briefly other ways of deriving quantum phase (super)spaces and we compare the spinorial Snyder type models defining bosonic or fermionic quantum-deformed spinors.From 2d droplets to 2d Yang-Millshttps://www.zbmath.org/1483.811002022-05-16T20:40:13.078697Z"Chattopadhyay, Arghya"https://www.zbmath.org/authors/?q=ai:chattopadhyay.arghya"Dutta, Suvankar"https://www.zbmath.org/authors/?q=ai:dutta.suvankar"Mukherjee, Debangshu"https://www.zbmath.org/authors/?q=ai:mukherjee.debangshu"Neetu"https://www.zbmath.org/authors/?q=ai:neetu.babbarSummary: We establish a connection between time evolution of free Fermi droplets and partition function of \textit{generalised q}-deformed Yang-Mills theories on Riemann surfaces. Classical phases of \((0 + 1)\) dimensional unitary matrix models can be characterised by free Fermi droplets in two dimensions. We quantise these droplets and find that the modes satisfy an abelian Kac-Moody algebra. The Hilbert spaces \(\mathcal{H}_+\) and \(\mathcal{H}_-\) associated with the upper and lower free Fermi surfaces of a droplet admit a Young diagram basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the large \(N\) limit, equal to the quadratic Casimir of \(u(N)\).
We establish an exact mapping between states in \(\mathcal{H}_\pm\) and geometries of droplets. In particular, coherent states in \(\mathcal{H}_\pm\) correspond to classical deformation of upper and lower Fermi surfaces. We prove that correlation between two coherent states in \(\mathcal{H}_\pm\) is equal to the chiral and anti-chiral partition function of \(2d\) Yang-Mills theory on a cylinder. Using the fact that the full Hilbert space \(\mathcal{H}_+ \otimes \mathcal{H}_-\) admits a \textit{composite} basis, we show that correlation between two classical droplet geometries is equal to the full \(U(N)\) Yang-Mills partition function on cylinder. We further establish a connection between higher point correlators in \(\mathcal{H}_\pm\) and higher point correlators in \(2d\) Yang-Mills on Riemann surface. There are special states in \(\mathcal{H}_\pm\) whose transition amplitudes are equal to the partition function of 2\textit{d q}-deformed Yang-Mills and in general character expansion of Villain action. We emphasise that the \(q\)-deformation in the Yang-Mills side is related to special deformation of droplet geometries without deforming the gauge group associated with the matrix model.Chern-Simons perturbative series revisitedhttps://www.zbmath.org/1483.811082022-05-16T20:40:13.078697Z"Lanina, E."https://www.zbmath.org/authors/?q=ai:lanina.elena|lanina.e-g"Sleptsov, A."https://www.zbmath.org/authors/?q=ai:sleptsov.alexey"Tselousov, N."https://www.zbmath.org/authors/?q=ai:tselousov.nSummary: A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with \(SU(N)\) gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra \(ZU(\mathfrak{sl}_N)\) is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved.Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structurehttps://www.zbmath.org/1483.811092022-05-16T20:40:13.078697Z"Lanina, E."https://www.zbmath.org/authors/?q=ai:lanina.e-g|lanina.elena"Sleptsov, A."https://www.zbmath.org/authors/?q=ai:sleptsov.alexey"Tselousov, N."https://www.zbmath.org/authors/?q=ai:tselousov.nSummary: We have recently proposed [Phys. Lett., B 823, Article ID 136727, 8 p. (2021; Zbl 1483.81108)] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with \(SU(N)\) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.
First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots \(T[2, 2 k + 1]\). Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank \(N\) and the representation \(R\).The Veneziano amplitude via mostly BRST exact operatorhttps://www.zbmath.org/1483.811192022-05-16T20:40:13.078697Z"Kishimoto, Isao"https://www.zbmath.org/authors/?q=ai:kishimoto.isao"Sasaki, Tomoko"https://www.zbmath.org/authors/?q=ai:sasaki.tomoko"Seki, Shigenori"https://www.zbmath.org/authors/?q=ai:seki.shigenori"Takahashi, Tomohiko"https://www.zbmath.org/authors/?q=ai:takahashi.tomohikoSummary: The Veneziano amplitude is derived from fixing one degree of freedom of \(PSL(2, \mathbb{R})\) symmetry by the insertion of a mostly BRST exact operator. Evaluating the five-point function which consists of four open string tachyons and this gauge fixing operator, we find it equals the Veneziano amplitude up to a sign factor. The sign factor is interpreted as a signed intersection number. The result implies that the mostly BRST exact operator, which is originally used to provide two-point string amplitudes, correctly fixes the \(PSL(2, \mathbb{R})\) gauge symmetry for general amplitudes. We conjecture an expression for general \(n\)-point tree amplitudes with an insertion of this gauge fixing operator.Kink solutions in logarithmic scalar field theory: excitation spectra, scattering, and decay of bionshttps://www.zbmath.org/1483.811252022-05-16T20:40:13.078697Z"Belendryasova, Ekaterina"https://www.zbmath.org/authors/?q=ai:belendryasova.ekaterina"Gani, Vakhid A."https://www.zbmath.org/authors/?q=ai:gani.vakhid-a"Zloshchastiev, Konstantin G."https://www.zbmath.org/authors/?q=ai:zloshchastiev.konstantin-gSummary: We consider the \((1 + 1)\)-dimensional Lorentz-symmetric field-theoretic model with logarithmic potential having a Mexican-hat form with two local minima similar to that of the quartic Higgs potential in conventional electroweak theory with spontaneous symmetry breaking and mass generation. We demonstrate that this model allows topological solutions -- kinks. We analyze the kink excitation spectrum, and show that it does not contain any vibrational modes. We also study the scattering dynamics of kinks for a wide range of initial velocities. The critical value of the initial velocity occurs in kink-antikink collisions, which thus differentiates two regimes. Below this value, we observe the capture of kinks and their fast annihilation; while above this value, the kinks bounce off and escape to spatial infinities. Numerical studies show no resonance phenomena in the kink-antikink scattering.Topological axion electrodynamics and 4-group symmetryhttps://www.zbmath.org/1483.811292022-05-16T20:40:13.078697Z"Hidaka, Yoshimasa"https://www.zbmath.org/authors/?q=ai:hidaka.yoshimasa"Nitta, Muneto"https://www.zbmath.org/authors/?q=ai:nitta.muneto"Yokokura, Ryo"https://www.zbmath.org/authors/?q=ai:yokokura.ryoSummary: We study higher-form symmetries and a higher group in the low energy limit of a \((3 + 1)\)-dimensional axion electrodynamics with a massive axion and a massive photon. A topological field theory describing topological excitations with the axion-photon coupling, which we call a topological axion electrodynamics, is obtained in the low energy limit. Higher-form symmetries of the topological axion electrodynamics are specified by equations of motion and Bianchi identities. We find that there are induced anyons on the intersections of symmetry generators. By a link of worldlines of the anyons, we show that the worldvolume of an axionic domain wall is topologically ordered. We further specify the underlying mathematical structure elegantly describing all salient features of the theory to be a 4-group.Quantisation of Lorentz invariant scalar field theory in non-commutative space-time and its consequencehttps://www.zbmath.org/1483.811312022-05-16T20:40:13.078697Z"Harikumar, E."https://www.zbmath.org/authors/?q=ai:harikumar.e"Rajagopal, Vishnu"https://www.zbmath.org/authors/?q=ai:rajagopal.vishnuSummary: Quantisation of Lorentz invariant scalar field theory in Doplicher-Fredenhagen-Roberts (DFR) space-time, a Lorentz invariant, non-commutative space-time is studied. We use an approach to quantisation that is based on the equations of motion alone and derive the equal time commutation relation between Doplicher-Fredenhagen-Roberts-Amorim (DFRA) scalar field and its conjugate, which has non-commutative dependent modifications, but the corresponding creation and annihilation operators obey usual algebra.
We show that imposing the condition that the commutation relation between the field and its conjugate is same as that in the commutative space-time leads to a deformation of the algebra of quantised oscillators. Both these deformed commutation relations derived are valid to all orders in the non-commutative parameter. By analysing the first non-vanishing terms which are \(\theta^3\) order, we show that the deformed commutation relations scale as \(1/\lambda^4\), where \(\lambda\) is the length scale set by the non-commutativity of the space-time. We also derive the conserved currents for DFRA scalar field. Further, we analyse the effects of non-commutativity on Unruh effect by analysing a detector coupled to the DFRA scalar field, showing that the Unruh temperature is not modified but the thermal radiation seen by the accelerated observer gets correction due to the non-commutativity of space-time.Lorentz-symmetry violation in the electroweak sector: scattering processes in future \(e^+ e^-\) collidershttps://www.zbmath.org/1483.811332022-05-16T20:40:13.078697Z"De Fabritiis, P."https://www.zbmath.org/authors/?q=ai:de-fabritiis.p"Malta, P. C."https://www.zbmath.org/authors/?q=ai:malta.p-c"Neves, M. J."https://www.zbmath.org/authors/?q=ai:neves.mario-junSummary: We study CPT-odd non-minimal Lorentz-symmetry violating couplings in the electroweak sector modifying the interactions between leptons, gauge mediators and the Higgs boson. The tree-level (differential) cross sections for three important electroweak processes are discussed: \(e^+ e^- \to ZH\), \(e^+ e^- \to Z Z\) and \(\gamma \gamma \to W^+ W^-\). By considering next-generation \(e^+ e^-\) colliders reaching center-of-mass energies at the TeV scale and the estimated improved precision for the measurements of the respective cross sections, we are able to project upper bounds on the purely time-like background 4-vector as strict as \(\lesssim 10^{-5}\) \(\mathrm{GeV}^{-1}\), in agreement with previous work on similar Lorentz-violating couplings.Spacetimes with continuous linear isotropies. II: Boostshttps://www.zbmath.org/1483.830192022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: Conditions are found which ensure that local boost invariance (LBI), invariance under a linear boost isotropy, implies local boost symmetry (LBS), i.e. the existence of a local group of motions such that for every point \(P\) in a neighbourhood there is a boost leaving \(P\) fixed. It is shown that for Petrov type D spacetimes this requires LBI of the Riemann tensor and its first derivative. That is also true for most conformally flat spacetimes, but those with Ricci tensors of Segre type [1(11,1)] may require LBI of the first three derivatives of curvature to ensure LBS.
For Parts I and III, see [the author, ibid. 53, No. 6, Paper No. 57, 21 p. (2021; Zbl 1483.83006); ibid. 53, No. 10, Paper No. 96, 22 p. (2021; Zbl 1483.83020)].Spacetimes with continuous linear isotropies. III: Null rotationshttps://www.zbmath.org/1483.830202022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: It is shown that in many of the possible cases local null rotation invariance of the curvature and its first derivatives is sufficient to ensure that there is an isometry group \(G_r\) with \(r\ge 3\) acting on (a neighbourhood of) the spacetime and containing a null rotation isotropy. The exceptions where invariance of the second derivatives is additionally required to ensure this conclusion are Petrov type N Einstein spacetimes, spacetimes containing ``pure radiation'' (a Ricci tensor of Segre type [(11,2)]), and conformally flat spacetimes with a Ricci tensor of Segre type [1(11,1)] (a ``tachyon fluid'').
For Parts I and II, see [the author, ibid. 53, No. 6, Paper No. 57, 21 p. (2021; Zbl 1483.83006); ibid. 53, No. 6, Paper No. 61, 12 p. (2021; Zbl 1483.83019)].Schwarzschild-like black hole with a topological defect in bumblebee gravityhttps://www.zbmath.org/1483.830462022-05-16T20:40:13.078697Z"Güllü, İbrahim"https://www.zbmath.org/authors/?q=ai:gullu.ibrahim"Övgün, Ali"https://www.zbmath.org/authors/?q=ai:ovgun.aliSummary: In this paper, we derive an exact black hole spacetime metric in the Einstein-Hilbert-Bumblebee (EHB) gravity around global monopole field. We study the horizon, temperature, and the photon sphere of the black hole. Using the null geodesics equation, we obtain the shadow cast by the Schwarzschild-like black hole with a topological defect in Bumblebee gravity. Interestingly, the radius of shadow of the black hole increases with increase in the global monopole parameter. We also visualize the shadows and energy emission rates for different values of parameters. Moreover, using the Gauss-Bonnet theorem, we calculate the deflection angle in weak field limits and we discuss the possibility of testing the effect of global monopole field and bumblebee field, on a weak deflection angle. We find that the global monopole parameter and also Lorentz symmetry breaking parameter has an increasing effect on the deflection angle.The effect of stationary axisymmetric spacetimes in interferometric visibilityhttps://www.zbmath.org/1483.850122022-05-16T20:40:13.078697Z"Basso, Marcos L. W."https://www.zbmath.org/authors/?q=ai:basso.marcos-l-w"Maziero, Jonas"https://www.zbmath.org/authors/?q=ai:maziero.jonasSummary: In this article, we consider a scenario in which a spin-1/2 quanton goes through a superposition of co-rotating and counter-rotating circular paths, which play the role of the paths of a Mach-Zehnder interferometer in a stationary and axisymmetric spacetime. Since the spin of the particle plays the role of a quantum clock, as the quanton moves in a superposed path it gets entangled with the momentum (or the path), and this will cause the interferometric visibility (or the internal quantum coherence) to drop, since, in stationary axisymmetric spacetimes there is a difference in proper time elapsed along the two trajectories. However, as we show here, the proper time of each path will couple to the corresponding local Wigner rotation, and the effect in the spin of the superposed particle will be a combination of both. Besides, we discuss a general framework to study the local Wigner rotations of spin-1/2 particles in general stationary axisymmetric spacetimes for circular orbits.Linear control systems on the homogeneous spaces of the 2D Lie grouphttps://www.zbmath.org/1483.930362022-05-16T20:40:13.078697Z"Ayala, Víctor"https://www.zbmath.org/authors/?q=ai:ayala.victor"Da Silva, Adriano"https://www.zbmath.org/authors/?q=ai:silva.adriano-da"Torreblanca, María"https://www.zbmath.org/authors/?q=ai:torreblanca.maria-luisaSummary: In this paper, we classify all the possible linear control systems on the homogeneous spaces of the 2D solvable Lie group and study their controllability and control sets.