Related papers: The Congruence Subgroup Problem for finitely gener…

Let $\Gamma < \mathrm{GL}_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure, $\mathrm{Sub}(\Gamma)$ the space of all subgroups of $\Gamma$ with the, compact, metric, Chabauty topology. An invariant…

A graph Gamma is said to be 2-arc-transitive if its full automorphism group Aut(\Gamma) has a single orbit on ordered paths of length 2, and for G\leq Aut(\Gamma), \Gamma is G-regular if G is regular on the vertex set of \Gamma. Let G be a…

Given a commensurated subgroup $\Lambda$ of a group $\Gamma$, we completely characterize when the inclusion $\Lambda\leq \Gamma$ is $C^*$-irreducible and provide new examples of such inclusions. In particular, we obtain that…

We study groups of reversible cellular automata, or CA groups, on groups. More generally, we consider automorphism groups of subshifts of finite type on groups. It is known that word problems of CA groups on virtually nilpotent groups are…

Suppose that $G$ is a finite group and $H$ is a nilpotent subgroup of $G$. If a character of $H$ induces an irreducible character of $G$, then the generalized Fitting subgroup of $G$ is nilpotent.

For a locally finite, connected graph $\Gamma$, let $\operatorname{Map}(\Gamma)$ denote the group of proper homotopy equivalences of $\Gamma$ up to proper homotopy. Excluding sporadic cases, we show $\operatorname{Aut}(S(M_\Gamma)) \cong…

We define the completion of an associative algebra $A$ in a set $M=\{M_1,\dots,M_r\}$ of $r$ right $A$-modules in such a way that if $\mathfrak a\subseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the (right) module…

We show that for a Hecke pair $(G, \Gamma)$ the $C^*$-completions $C^*(L^1(G, \Gamma))$ and $pC^*(\bar{G})p$ of its Hecke algebra coincide whenever the group algebra $L^1(\bar{G})$ satisfies a spectral property which we call…

Let $\Gamma$ be the random bipartite graph, a countable graph with two infinite sides, edges randomly distributed between the sides, but no edges within a side. In this paper, we investigate the reducts of $\Gamma$ that preserve sides. We…

Let $\Gamma$ be a finitely generated torsion free nilpotent group, and let $A^\omega$ be the space of infinite words over a finite alphabet $A$. We investigate two types of self-similar actions of $\Gamma$ on $A^\omega$, namely the…

Let $\Gamma$ be a finitely generated group which is hyperbolic relative to a finite family $\{H_1,...,H_n\}$ of subgroups. We prove that $\Gamma$ is uniformly embeddable in a Hilbert space if and only if each subgroup $H_i$ is uniformly…

The complex algebra of an inverse semigroup with finitely many idempotents in each $\mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using $C^*$-algebras for inverse semigroups satisfying this…

We study the complexity of constraint satisfaction problems for templates $\Gamma$ that are first-order definable in $(\Bbb Z; succ)$, the integers with the successor relation. Assuming a widely believed conjecture from finite domain…

If G is a non-nilpotent group and nil(G) = {g \in G : <g, h> is nilpotent for all h\in G}, the nilpotent graph of G is the graph with set of vertices G-nil(G) in which two distinct vertices are related if they generate a nilpotent subgroup…

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order…

Let $M$ be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes_\sigma \Gamma$. When $\Gamma$ is amenable, we obtain a complete…

We define several "standard" subgroups of the automorphism group Aut(G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut(G). If C is the commutation graph of G, we show…

Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \leq K \leq -1$. There exist a positive constant $C(n,a)$ such that for any finitely generated discrete group $\Gamma$ acting on $X$, then either…

The notion of the quantum automorphism group of a graph was introduced by J. Bichon in 2003 and T. Banica in 2005 respectively. This article explores primarily the quantum automorphism group of a graph $\Gamma$, denoted by…

A graph $\Gamma$ is $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such…