Related papers: Concise formulae in groups of non-positive curvatu…

A word $w$ is concise in a class of groups $\mathcal{C}$ if, for every group $G$ in $\mathcal{C}$, the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in $G$. This notion can be naturally extended to…

In this paper we initiate a study of first-order rich groups, i.e., groups where the first-order logic has the same power as the weak second order logic. Surprisingly, there are quite a lot of finitely generated rich groups, they are…

In this note, we show that the satisfiability of equations and inequations with recognisable constraints is decidable in groups that are virtually direct products of finitely many hyperbolic groups.

We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $\delta$-hyperbolic spaces with general type factors and associated subdirect products.…

We show that the group $\langle a,b,c,t : a^t=b,b^t=c,c^t=ca^{-1} \rangle$ is profinitely rigid amongst free-by-cyclic groups, providing the first example of a hyperbolic free-by-cyclic group with this property.

We give a new proof that free Burnside groups of sufficiently large even exponents are infinite. The method is very flexible and can also be used to study (partially) periodic quotients of any group which admits an action on a hyperbolic…

In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more…

This is an announcement of some of the results obtained as a part of the second author's Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is…

We show that Morse elements are generic in acylindrically hyperbolic groups. As an application, we observe that fully irreducible outer automorphisms are generic in the outer automorphism group of a finite-rank free group.

We prove that every finitely presented group with positive first $\ell^2$-Betti number that virtually surjects onto $\mathbb Z$ is acylindrically hyperbolic. In particular, this implies acylindrical hyperbolicity of finitely presented…

We show that the universal theory of torsion groups is strongly contained in the universal theory of finite groups. This answers a question of Dyson. We also prove that the universal theory of some natural classes of torsion groups is…

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a…

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We…

We study a class of two-generator two-relator groups, denoted $J_n(m,k)$, that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature…

We describe a procedure which verifies that a group given by generators and relators is word-hyperbolic. This procedure always works with a group which is word-hyperbolic, provided there is sufficient memory and time devoted to the problem.…

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…

We provide an analogue of Strebel's classification of geodesic triangles in classical $C'(\frac16)$ groups for groups given by Wise's cubical presentations satisfying sufficiently strong metric cubical small cancellation conditions. Using…

We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an acylindrically hyperbolic group and $E(G)$ denotes the unique…

We consider groups defined by cyclic presentations where the defining word has length three and the cyclic presentation satisfies the T(6) small cancellation condition. We classify when these groups are hyperbolic. When combined with known…

Presentations for the holomorphs of abelian groups of the form $C_{p^n} \times 1^{m}$ for $p$=2 or an odd prime are given. These presentations extend the results given in Burnside's well-known text on finite groups on the holomorphs for the…