Related papers: Non-residually finite extensions of arithmetic gro…

We extend to the context of algebraic groups a classic result on extensions of abstract groups relating the set of isomorphism classes of extensions of $G$ by $H$ with that of extensions of $G$ by the center $Z$ of $H$. The proof should be…

If $g\in G$ is a non-trivial element in a residually finite group, then there exists by definition a finite group $Q$ and a homomorphism $\varphi: G \to Q$ such that $\varphi(g) \neq e$. The residual finiteness growth $\text{RF}_G$ of a…

In a series of recent contributions on the notion of global breadth $\mathbf{B}(G)$ of a finite group $G$, it was interesting to observe the structural conditions arising from the classification of finite groups of $\mathbf{B}(G)=8$. This…

A countable group is residually finite if every nontrivial element can act nontrivially on a finite set. When a group fails to be residually finite, we might want to measure how drastically it fails - it could be that only finitely many…

A group $G$ is called residually finite if for every non-trivial element $g \in G$, there exists a finite quotient $Q$ of $G$ such that the element $g$ is non-trivial in the quotient as well. Instead of just investigating whether a group…

We construct an infinite finitely generated recursively presented residually finite algorithmically finite group $G$ answering thereby a question of Myasnikov and Osin. Moreover, $G$ is "very infinite" and "very algorithmically finite" in…

In this article we study a class of central extensions of $\mathbb{Z}\wr\mathbb{Z}$, as first described by Hall. On the one hand, we consider groups of this type with cyclic centre, our construction yields a rich class of groups. In…

We classify finite groups $G$, such that the group algebra, $\mathbb{Q}G$ (over the field of rational numbers $\mathbb{Q}$), is the direct product of the group algebra $\mathbb{Q}[G/N]$ of a proper factor group $G/N$, and some division…

This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group G, Parry showed how to define a G-extension S_A from a square matrix A over Z_+G,…

Let G be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if G has the congruence subgroup property, then the number of n-dimensional irreducible representations of G grows like n^a, where a is a…

Given a finitely generated residually finite group $G$, the residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ bounds the size of a finite group $Q$ needed to detect an element of norm at most $r$. More specifically, if…

A subset S of a group G invariably generates G if G = <s^(g(s)) | s in S> for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a…

We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism.

In this note, we compute the {\Sigma}^1(G) invariant when 1 {\to} H {\to} G {\to} K {\to} 1 is a short exact sequence of finitely generated groups with K finite. As an application, we construct a group F semidirect Z_2 where F is the R.…

A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and…

We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices $\Gamma < \mathrm{PU}(n,1)$ simple type. We prove that the preimage of $\Gamma$ in any connected cover of $\mathrm{PU}(n,1)$, in particular the…

For various nonsolvable groups $G$, we prove the existence of extensions of the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set for $G$.…

We show that many $2$-dimensional Artin groups are residually finite. This includes $3$-generator Artin groups with labels $\geq 4$ except for $(2m+1, 4,4)$ for any $m\geq 2$. As a first step towards residual finiteness we show that these…

The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying…

We provide new examples of groups without rational cross-sections (also called regular normal forms), using connections with bounded generation and rational orders on groups. Specifically, our examples are extensions of infinite torsion…