Related papers: Arithmetic groups with isomorphic finite quotients

We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the…

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite…

We determine the finite groups whose real irreducible representations have different degrees.

For a number field K, we show that any S-arithmetic subgroup of SL_2(K) contains a subgroup of finite index generated by three elements if card(S)> 1.

In this article, I study some classes of finitely presented groups with the aim of finding out whether the maximal metabelian quotients of the members of these classes admit finite presentations. The considered classes include those of…

We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group: We establish a semidirect product…

We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type FP_\infty by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups.

Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$,…

It is well known that if two finite groups have the same symmetric tensor categories of representations over C, then they are isomorphic. We study the following question: when do two finite groups G1,G2 have the same tensor categories of…

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C_1,...,C_n are finite cyclic groups then there exists a finite group G which is…

We prove two results. (1) There is an absolute constant $D$ such that for any finite quasisimple group $S$, given 2D arbitrary automorphisms of $S$, every element of $S$ is equal to a product of $D$ `twisted commutators' defined by the…

Answering a question of Dan Haran and generalizing some results of Aschbacher-Guralnick and Suzuki, we prove that given a set of primes pi, any finite group can be generated by a pi-subgroup and a pi'-subgroup. This gives a free product…

We consider the fundamental group $\pi$ of a surface of finite type equipped with the infinite generating set consisting of all simple closed curves. We show that every nilpotent quotient of $\pi$ has finite diameter with respect to the…

Let G be a simple finite graph such that each vertex has an integer value and different vertices have different values. Let S be a finite non-empty set of primes. We call G an S-graph if any two vertices are connected by an edge if and only…

This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable…

Let $\mathbb F$ be an algebraically closed field, $G$ be an abelian group, and let $A$ and $B$ be arbitrary finite-dimensional $G$-graded simple algebras over $\mathbb F$. We prove that $A$ and $B$ are isomorphic if, and only if, they…

Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $\phi \in \text{Aut}(G)$ such that $\phi(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. This result leads to our development of…

For affine symmetric groups we construct finite $W$-graphs corresponding to two-row shapes, and prove their uniqueness. This gives the first non-trivial family of examples of finite $W$-graphs in an affine type. We compare our construction…

We show that any group with arbitrarily large finite quotients admits generating sets with respect to which it has arbitrarily large finite dead-end depth. This extends a joint result with Riley and partially answers a question asked there.

We show that property (T) is not profinite, that is, we construct two finitely generated residually finite groups which have isomorphic profinite completions while one admits property (T) and the other does not. This settles a question…