Related papers: Rigidity and flexibility results for groups with a…

The paper investigates two invariants for totally disconnected locally compact groups: the number of ends and the rational discrete cohomological dimension. For such a compactly generated group $G$ it is shown that its number of ends can be…

The weak regular coherence is a coarse property of a finitely generated group $\Gamma$. It was introduced by G. Carlsson and this author to play the role of a weakening of Waldhausen's regular coherence as part of computation of the…

The nonsoluble length $\lambda(G)$ of a finite group $G$ is defined as the minimum number of nonsoluble factors in a normal series of $G$ each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The…

Let K be a number field, let A be a finite dimensional semisimple K-algebra and let Lambda be an O_K-order in A. It was shown in previous work that, under certain hypotheses on A, there exists an algorithm that for a given (left)…

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x in G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite…

Let $G$ be a countable group. We introduce several equivalence relations on the set ${\rm Sub}(G)$ of subgroups of $G$, defined by properties of the quasi-regular representations $\lambda_{G/H}$ associated to $H\in {\rm Sub}(G)$ and compare…

A finite simple graph $\Gamma$ determines a quotient $P_\Gamma$ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior…

Let $G$ be a simply connected, solvable Lie group and $\Gamma$ a lattice in $G$. The deformation space $\mathcal{D}(\Gamma,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(\Gamma,G)$ of all lattice…

Let $\Gamma$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $\Gamma_0$ has critical exponent equal to the critical exponent of $\Gamma$ if and only if $\Gamma /…

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an…

We discuss questions by Mostow \cite{Mo1}, Bers \cite{B} and Krushkal \cite{Kr1, Kr2} about uniqueness of a conformal or spherical CR structure on the sphere at infinity $\partial H_\mathbb{F}^n$ of symmetric rank one space $H_\mathbb{F}^n$…

We study the rigidity in the sense of Zimmer for higher rank lattice actions on dendrites and show that: (1) if $\Gamma$ is a higher rank lattice and $X$ is a nondegenerate dendrite with no infinite order points, then any action of $\Gamma$…

Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the non-identity elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. Let $G$ be a 2-generated…

We prove that for lambda = beta_omega or just lambda strong limit singular of cofinality aleph_0, if there is a universal member in the class K^lf_lambda of locally finite groups of cardinality lambda, then there is a canonical one…

Let $\sigma=\{\sigma_j\,:\, j\in J\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A subgroup $X$ of a finite group $G$ is~\textit{$\sigma$-subnormal} in $G$ if there exists a chain of subgroups $$X=X_0\leq X_1\leq\ldots\leq…

H.S.M. Coxeter showed that a group $\Gamma$ is a finite reflection group of an Euclidean space if and only if $\Gamma$ is a finite Coxeter group. In this paper, we define {\it reflections} of geodesic spaces in general, and we prove that…

Let $\Gamma$ be a countable discrete group. We say that $\Gamma$ has $C^*$-invariant subalgebra rigidity (ISR) property if every $\Gamma$-invariant $C^*$-subalgebra $\mathcal{A}\le C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for some normal…

We single out a large class of groups ${\mathscr{M}}$ for which the following unique prime factorization result holds: if $\Gamma_1,\dots,\Gamma_n\in {\mathscr{M}}$ and $\Gamma_1\times\dots\times\Gamma_n$ is measure equivalent to a product…

Let $M$ be a compact, connected manifold of positive dimension and let $\mathcal G\leq\textrm{Homeo}(M)$ be \emph{locally approximating} in the sense that for all open $U\subseteq M$ compactly contained in a single Euclidean chart of $M$,…

We establish rigidity theorems for graph product von Neumann algebras $M_\Gamma=*_{v,\Gamma}M_v$ associated to finite simple graphs $\Gamma$ and families of tracial von Neumann algebras $(M_v)_{v\in\Gamma}$. We consider the following three…