Related papers: Embedding hyperbolic groups into finitely presente…

We prove that every finitely presentable group G arises as the fundamental group of an orientable 3-complex obtained from a hyperbolic link complement, by coning each boundary torus of the link exterior to a distinct point. We define the…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively…

We consider the class of groups whose word problem is poly-context-free; that is, an intersection of finitely many context-free languages. We show that any group which is virtually a finitely generated subgroup of a direct product of free…

A finitely generated group is lacunary hyperbolic if one of its asymptotic cones is an $\mathbb{R}$-tree. In this article we give a necessary and sufficient condition on lacunary hyperbolic groups in order to be stable under free product by…

For a hierarchically hyperbolic group, we provide sufficient conditions under which suitable powers of a finite collection of elements generate a right-angled Artin subgroup. Under additional hypotheses, we further show that this subgroup…

This paper is a survey of results proved in recent years that pertain to classifying cobounded hyperbolic actions of any group $G$. In other words, we discuss results that allow us to describe the partially ordered set $\mathcal{H}(G)$,…

We prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group $nV$. It follows that many other groups can be embedded into some $nV$ (e.g., any finite extension of any of Haglund and Wise's…

We present novel constructions concerning the homology of finitely generated groups. Each construction draws on ideas of Gilbert Baumslag. There is a finitely presented acyclic group $U$ such that $U$ has no proper subgroups of finite index…

Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb,…

Every nonabelian finite simple group of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, has a presentation with a bounded number of generators and relations and total length $O(\log n…

We show that properties $F_n$ and $FP_n$ hold for a relatively hyperbolic group if and only if they hold for all the peripheral subgroups. As an application we show that there are at least countably many distinct quasi-isometry classes of…

The Diophantine problem for a monoid $M$ is the decision problem to decide whether any given system of equations has a solution in $M$. In this note, we give a simple example of a context-free, word-hyperbolic, finitely presented, special…

We prove that the only finite factor-representations of the Higman-Thompson groups $\{F_{n,r}\}$, $ \{G_{n,r}\}$ are the regular representations and scalar representations arising from group abelianizations. As a corollary, we obtain that…

We generalise the constructions of Brady and Lodha to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type $F_3$. By calculating the Euler characteristic of the hyperbolic groups…

Let $w$ be a group-word. Suppose that the set of all $w$-values in a profinite group $G$ is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup $w(G)$ depends on the…

We prove that acylindrically hyperbolic groups are monotileable. That is, every finite subset of the group is contained in a finite tile. This provides many new examples of monotileable groups, and progress on the question of whether every…

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is…

Over each nontrivial finite group $G$, there exists a finite system of equations having no solutions in larger finite groups but having a solution in a periodic group containing $G$. We prove several similar facts about amenable, orderable,…

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a…

A finitely presented group is semistable at infinity if all proper rays in the Cayley 2-complex are properly homotopic. A long standing open question asks whether all finitely presented groups are semistable at infinity. This article…