Mathematics - Group Theory Publications (50)

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Mathematics - Group Theory Publications

We present a survey of results on profinite semigroups and their link with symbolic dynamics. We develop a series of results, mostly due to Almeida and Costa and we also include some original results on the Sch\"utzenberger groups associated to a uniformly recurrent set. Read More


We show that the level sets of automorphisms of free groups with respect to the Lipschitz metric are connected as subsets of Culler-Vogtmann space. In fact we prove our result in a more general setting of deformation spaces. As applications, we give metric solutions of the conjugacy problem for irreducible automorphisms and the detection of reducibility. Read More


We establish a lower bound on the spectral gap of the Laplace operator on special linear groups using conic optimisation. In particular, this provides a constructive (but computer assisted) proof that these groups have Kazhdan property (T). A software for such optimisation for other finitely presented groups is provided. Read More


A group is $\frac{3}{2}$-generated if every non-identity element is contained in a generating pair. A conjecture of Breuer, Guralnick and Kantor from 2008 asserts that a finite group is $\frac{3}{2}$-generated if and only if every proper quotient of the group is cyclic, and recent work of Guralnick reduces this conjecture to almost simple groups. In this paper, we prove a stronger from of the conjecture for almost simple symplectic and odd-dimensional orthogonal groups. Read More


This is an expository article on properties of actions on Lie groups by subgroups of their automorphism groups. After recalling various results on the structure of the automorphism groups, we discuss actions with dense orbits, invariant and quasi-invariant measures, the induced actions on the spaces of probability measures on the groups, and results concerning various issues in ergodic theory, topological dynamics, smooth dynamical systems, and probability theory on Lie groups. Read More


We show that the simple elements of the dual Garside structure of an Artin group of type $D_n$ are Mikado braids, giving a positive answer to a conjecture of Digne and the second author. To this end, we use an embedding of the Artin group of type $D_n$ in a suitable quotient of an Artin group of type $B_n$ noticed by Allcock, of which we give a simple algebraic proof here. This allows one to give a characterization of the Mikado braids of type $D_n$ in terms of those of type $B_n$ and also to describe them topologically. Read More


We prove the orderability of the Witzel-Zaremsky-Thompson group for a direct system of orderable groups under a certain compatibility assumption. Read More


We prove that the set of symmetrized conjugacy classes of the kernel of the Calabi homomorphism on the group of area-preserving diffeomorphisms of the $2$-disk is not quasi-isometric to the half line. Read More


For any finite group $G$, a natural question to ask is the order of the smallest possible automorphism group for a Cayley graph on $G$. A particular Cayley graph whose automorphism group has this order is referred to as an MRR (Most Rigid Representation), and its Cayley index is a numerical indicator of this value. Study of GRRs showed that with the exception of two infinite families and seven individual groups, every group admits a Cayley graph whose MRR is a GRR, so that the Cayley index is 1. Read More


For finitely generated groups $H$ and $G$, we observe that $H$ admits a translation-like action on $G$ implies there is a regular map, which was introduced in Benjamini, Schramm and Tim\'{a}r's joint paper, from $H$ to $G$. Combining with several known obstructions to the existence of regular maps, we have various applications. For example, we show that the Baumslag-Solitar groups do not admit translation-like actions on the classical lamplighter group. Read More


We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We prove more generally that this result holds for all products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa's class S. We deduce a W* strong rigidity theorem for irreducible actions of such product groups and we prove strong solidity of the associated locally compact group von Neumann algebras. Read More


We consider $f, h$ homeomorphims generating a faithful $BS(1,n)$-action on a closed surface $S$, that is, $h f h^{-1} = f^n$, for some $ n\geq 2$. According to \cite{GL}, after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\Lambda$ of the action, included in $Fix(f)$. Here, we suppose that $f$ and $h$ are $C^1$ in neighbourhood of $\Lambda$ and any point $x\in\Lambda$ admits an $h$-unstable manifold $W^u(x)$. Read More


Using the work of Behrstock, Hagen, and Sisto on contact graphs for $CAT(0)$ cubical groups, we define loxodromic elements and purely loxodromic subgroups in right-angled Coxeter groups. We prove that finitely generated purely loxodromic subgroups of a right-angled Coxeter group fulfill several equivalent conditions that parallel characterizations of subgroups with the same name in a right-angled Artin group. We compare purely loxodromic subgroups in right-angled Coxeter groups with special subgroups, star-free subgroups, and stable subgroups. Read More


In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. Read More


We study the normal closure of a big power of one or several Dehn twists in a Mapping Class Group. We prove that it has a presentation whose relators consists only of commutators between twists of disjoint support, thus answering a question of Ivanov. Our method is to use the theory of projection complexes of Bestvina Bromberg and Fujiwara, together with the theory of rotating families, simultaneously on several spaces. Read More


Wreath products of non-discrete locally compact groups are usually not locally compact groups, nor even topological groups. We introduce a natural extension of the wreath product construction to the setting of locally compact groups. Read More


Here we define the notion of residually null sets, a $\sigma$-ideal of subsets of a Polish space which contains the meager sets and, in some contexts, generalizes the notion of universally null. From this $\sigma$-ideal we realize a $\sigma$-algebra of sets which is consistently a strict extension of the Baire property algebra. We then uncover a generalization to Pettis' Theorem which furnishes a new automatic continuity result. Read More


The spectrum of $L^2$ on a pseudo-unitary group $U(p,q)$ (we assume $p\ge q$ naturally splits into $q+1$ types. We write explicitly orthogonal projectors in $L^2$ to subspaces with uniform spectra (this is an old question formulated by Gelfand and Gindikin). We also write two finer separations of $L^2$. Read More


Let $G$, $N$ be groups and let $\psi \colon G \to Out(N)$ be a homomorphism. An extension for these data is a group $E$ which fits into the short exact sequence $1 \to N \to E \to G \to 1$ such that the action of $E$ on $N$ by conjugation induces $\psi$. It is classical that extensions can be characterised using ordinary group cohomology in dimension $2$ and $3$. Read More


We construct a linear system non-local game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed non-local game provides another counterexample to the "middle" Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). Read More


We obtain some structural properties of a factorised group $G = AB$, given that the conjugacy class sizes of certain elements in $A\cup B$ are not divisible by $p^2$, for some prime $p$. The case when $G = AB$ is a mutually permutable product is especially considered. Read More


This paper presents a description of the fourth dimension quotient, using the theory of limits of functors from the category of free presentations of a given group to the category of abelian groups. A functorial description of a quotient of the third Fox subgroup is given and, as a consequence, an identification (not involving an isolator) of the third Fox subgroup is obtained. It is shown that the limit over the category of free representations of the third Fox quotient represents the composite of two derived quadratic functors. Read More


We prove that coarse equivalence implies measure equivalence within the class of unimodular, amenable, locally compact, second countable, groups. For compactly generated groups within the aforementioned class, this shows that quasi-isometry implies measure equivalence. Read More


A Cayley graph for a group $G$ is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of $G$ is an element of the normaliser of $G$. A group $G$ is then said to be CCA if every connected Cayley graph on $G$ is CCA. We show that a finite simple group is CCA if and only if it has no element of order 4. Read More


It is well known that a dense subgroup $G$ of the complex unitary group $U(d)$ cannot be amenable as a discrete group when $d>1$. When $d$ is large enough we give quantitative versions of this phenomenon in connection with certain estimates of random Fourier series on the compact group $\bar G$ that is the closure of $G$. Roughly, we show that if $\bar G$ covers a large enough part of $U(d)$ in the sense of metric entropy then $G$ cannot be amenable. Read More


The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $\Gamma$ be a group and $\Delta$ a subgroup of finite index. Read More


We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. Read More


We provide a nilpotency criterion for fusion systems in terms of the vanishing of its cohomology with twisted coefficients. Read More


Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e. Read More


We consider a group-theoretic analogue of the classic subset sum problem. It is known that every virtually nilpotent group has polynomial time decidable subset sum problem. In this paper we use subgroup distortion to show that every polycyclic non-virtually-nilpotent group has NP-complete subset sum problem. Read More


We extend the Weil representation of infinite-dimensional symplectic group to a representation a certain category of linear relations. Read More


We explain why semistability of a one-ended proper CAT(0) space can be determined by the geodesic rays. This is applied to boundaries of CAT(0) groups. Read More


We study finite groups $G$ having a subgroup $H$ and $D \subset G \setminus H$ such that the multiset $\{ xy^{-1}:x,y \in D\}$ has every non-identity element occur the same number of times (such a $D$ is called a {\it difference set}). We show that $H$ has to be normal, that $|G|=|H|^2$, and that $|D \cap Hg|=|H|/2$ for all $g \notin H$. We show that $H$ is contained in every normal subgroup of prime index, and other properties. Read More


We investigate the relationship between endomorphisms of the Cuntz algebra ${\mathcal O}_2$ and endomorphisms of the Thompson groups $F$, $T$ and $V$ represented inside the unitary group of ${\mathcal O}_2$. For an endomorphism $\lambda_u$ of ${\mathcal O}_2$, we show that $\lambda_u(V)\subseteq V$ if and only if $u\in V$. If $\lambda_u$ is an automorphism of ${\mathcal O}_2$ then $u\in V$ is equivalent to $\lambda_u(F)\subseteq V$. Read More


We show that the class of finite groups with n automorphisms has the amalgamation property for every n. Consequently, the automorphism group of Philip Hall's universal locally finite group has ample generics, that is, it admits comeager diagonal conjugacy classes in all dimensions. Read More


In this paper, we continue the work begun by Cheng et al.~on classifying which of the $(n,k)$-star graphs are Cayley. We present a conjecture for the complete classification, and prove an asymptotic version of the conjecture, that is, the conjecture is true for all $k\geq 2$ when $n$ is sufficiently large. Read More


The concept of a C-approximable group, as introduced by Holt and Rees for a class of finite groups C, is a common generalization of the concepts of sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. Read More


We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$, a baric structure on $\mathcal{K}$. Read More


Let $G$ be either the Grigorchuk $2$-group or one of the Gupta-Sidki $p$-groups. We give new upper bounds for the diameters of the quotients of $G$ by its level stabilisers, as well as other natural sequences of finite-index normal subgroups. Our bounds are independent of the generating set, and are polylogarithmic functions of the group order, with explicit degree. Read More


This is a translation from Russian of the article by A.A.Vinogradov: On the free product of ordered groups, Mat. Read More


Subset currents on hyperbolic groups were introduced by Kapovich and Nagnibeda as a generalization of geodesic currents on hyperbolic groups, which were introduced by Bonahon and have been successfully studied in the case of the fundamental group $\pi_1 (\Sigma)$ of a compact hyperbolic surface $\Sigma$. Kapovich and Nagnibeda particularly studied subset currents on free groups. In this article, we develop the theory of subset currents on $\pi_1(\Sigma )$, which we call subset currents on $\Sigma$. Read More


A finitely generated subgroup H of a torsion-free hyperbolic group G is called immutable if there are only finitely many conjugacy classes of injections of H into G. We show that there is no uniform algorithm to recognize immutability, answering a uniform version of a question asked by the authors. Read More


We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of non positive curvature and which do not hold for arbitrary characteristic zero linear groups. In particular if such a linear group is finitely generated then centralisers virtually split and all finitely generated abelian subgroups are undistorted. If further the group is virtually torsion free (which always holds in characteristic zero) then we have a strong property on small subgroups: any subgroup either contains a non abelian free group or is finitely generated and virtually abelian, hence also undistorted. Read More


We apply our theory of partial flag spaces developed in previous articles to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti-Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl-Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces. Read More


In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. Read More


The fixed point building of a polarity of a Moufang quadrangle of type $F_4$ is a Moufang set, as is the fixed point building of a semi-linear automorphism of order $2$ of a Moufang octagon that stabilizes at least two panels of one type but none of the other. We show that these two classes of Moufang sets are, in fact, the same, that each member of this class can be constructed as the fixed point building of a group of order $4$ acting on a building of type $F_4$ and that the group generated by all the root groups of any one of these Moufang sets is simple. Read More


We consider low dimensional diffuse Bieberbach groups. In particular we classify diffuse Bieberbach groups up to dimension 6. We also answer a question from [S. Read More


Kazhdan constants of discrete groups are hard to compute and the actual constants are known only for several classes of groups. By solving a semidefinite programming problem by a computer, we obtain a lower bound of the Kazhdan constant of a discrete group. Positive lower bounds imply that the group has property (T). Read More


We define a notion of Euler totient for any irreducible subfactor planar algebra, using the M\"obius function of the biprojection lattice. We prove that if it is nonzero then there is a minimal 2-box projection generating the identity biprojection. We deduce a bridge between combinatorics and representations in finite groups theory. Read More