Related papers: A finitely generated branch group of exponential g…

Given a group word $w$ and a group $G$, the set of $w$-values in $G$ is denoted by $G_w$ and the verbal subgroup $w(G)$ is the one generated by $G_w$. In the present paper we consider profinite groups admitting a word $w$ such that the…

We prove the following version of Milnor's theorem on solvable groups of exponential growth: A finitely generated solvable group which is not polycyclic contains an ascending HNN extension. Consequently, a finitely generated solvable group…

We prove that every non-elementary hyperbolic group $G$ acts with maximal growth on some set $X$ such that every orbit of any element $g \in G$ is finite. As a side-product of our approach we prove that if $G$ is non-elementary hyperbolic,…

Let $m,n$ be positive integers and $p$ a prime. We denote by $\nu(G)$ an extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is a residually finite group satisfying some non-trivial identity $f…

Given a group $G$ with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of $G$ is finitely generated and virtually abelian of rank at most $2$. In particular this gives a new proof of the above…

In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group $G$ lying in a variety ${\mathcal M}$ can be embedded in a $4$-generated group $H \in…

A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend…

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…

If a finitely generated torsion free group K has the property that all finitely generated subgroups S of K are either small or have growth constant bounded uniformly away from 1 then a non proper HNN extension G of K, that is a semidirect…

We announce the folowing result: Any finitely generated non virtually solvable linear group over a field of characteristic zero has uniform exponential growth.

We prove that for every $n \in \mathbb{N}$ and $\delta>0$ there exists a word $w_n \in F_2$ of length $n^{2/3} \log(n)^{3+\delta}$ which is a law for every finite group of order at most $n$. This improves upon the main result of [A. Thom,…

We prove that there exists a finitely generated group that satisfies a group law with probability 1 but does not satisfy any group law. More precisely, we construct a finitely generated group G in which the probability that a random element…

We study fibers of word maps in finite, profinite, and residually finite groups. Our main result is that, for any word w in the free group on d generators, there exists $\epsilon > 0$ such that if G is a residually finite group with…

We show that free Burnside groups of sufficiently large odd exponent are non--amenable in a certain strong sense, more precisely, their left regular representations are isolated from the trivial representation uniformly on finite generating…

We introduce a new invariant of finitely generated groups, the ambiguity function, and prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the…

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Motivated by the study of Hausdorff dimension, we show that finitely generated closed subgroups $H$ of infinite index in $G$ never contain any infinite subgroups $K$…

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…

Let $F$ be a finitely generated non-abelian free group and $Q$ a finite quotient. Denote by $L_Q$ the language obtained by adding unary predicates $P_q$, $q\in Q$ to the language of groups. Using a slight generalization of some of the…

Given an abstract group $G$, we study the function $ab_n(G) := \sup_{|G:H| \leq n} |H/[H,H]|$. If $G$ has no abelian composition factors, then $ab_n(G)$ is bounded by a polynomial: as a consequence, we find a sharp upper bound for the…

There are several results in the literature concerning $p$-groups $G$ with a maximal elementary abelian normal subgroup of rank $k$ due to Thompson, Mann and others. Following an idea of Sambale we obtain bounds for the number of generators…