Related papers: One Variable Equations in Torsion-Free Hyperbolic …

We show that every finitely generated residually finite torsion group $G$ embeds in a finitely generated torsion group $\Gamma$ that is residually finite simple. In particular we show the existence of finitely generated infinite torsion…

We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on…

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…

Let $\Gamma$ be a discrete and torsion-free subgroup of $\mathrm{PU}(n,1)$, the group of biholomorphisms of the unit ball in $\mathbb{C}^{n}$, denoted by $\mathbb{H}^{n}_{\mathbb{C}}$. We show that if $\Gamma$ is Abelian, then…

Bestvina and Mess [BM] proved a remarkable formula for torsion free hyperbolic groups $$ \dim_L\partial\Gamma=cd_L\Gamma-1 $$ connecting the cohomological dimension of a group $\Gamma$ with the cohomological dimension of its boundary…

Let $\Gamma$ be a discrete group of isometries acting on the complex hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$. In this note, we prove that if $\Gamma$ is convex-cocompact, torsion-free, and the critical exponent $\delta(\Gamma)$ is…

Let $G$ be a non-trivial torsion free group and $s(t)=g_{1}t^{\epsilon_{1}}g_{2}t^{\epsilon_{2}} \cdots g_{n}t^{\epsilon_{n}}=1 \; (g_{i} \in G,\ \epsilon_i=\pm 1)$ be an equation over $G$ containing no blocks of the form…

We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of…

We give a necessary and sufficient condition for the fundamental group of a finite graph of groups with infinite cyclic edge groups to be acylindrically hyperbolic, from which it follows that a finitely generated group splitting over Z…

It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of…

We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively…

In this work we investigate tensor completions of groups by associative rings, which were introduced by R.Lyndon and G.Baumslag in 1960s. The main result states that there exists an algorithm that decides if a given finite system of…

We prove that an automorphism $\phi:F\to F$ of a finitely generated free group $F$ is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes.

A finitely generated subgroup H of a torsion-free hyperbolic group G is called immutable if there are only finitely many conjugacy classes of injections of H into G. We show that there is no uniform algorithm to recognize immutability,…

We show that any nonabelian free group $F$ of finite rank is homogeneous; that is for any tuples $\bar a$, $\bar b \in F^n$, having the same complete $n$-type, there exists an automorphism of $F$ which sends $\bar a$ to $\bar b$. We further…

This paper is the first in a sequence on the structure of sets of solutions to systems of equations over a free associative algebra. We start by constructing a Makanin-Razborov diagram that encodes all the homogeneous solutions to a…

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

We prove that if $G$ is a free-torsion group and $w(t)$ is a word in the alphabet $G \sqcup \{t^{\pm 1}\}$ with exponent sum one, then the group $<G,t|(w(t))^k = 1>$, where $k \geq 2$, is relatively hyperbolic with respect to $G$.

This paper studies automorphisms and monomorphisms of direct products $\Gamma=\Gamma_1\times\cdots\times\Gamma_r$ of finitely generated virtually solvable minimax groups, a class containing all virtually polycyclic groups. Under an…

Let $G$ be the group of automorphisms of a free group $F_\infty$ of infinite order. Let $H$ be the stabilizer of first $m$ generators of $F_\infty$. We show that the double cosets of $\Gamma$ with respect to $H$ admit a natural semigroup…