Related papers: Ascending Chains of Free Quasiconvex Subgroups

The flat-rank of a totally disconnected, locally compact group G is an integer, which is an invariant of G as a topological group. We generalize the concept of hyperbolic groups to the topological context and show that a totally…

In this paper we provide new examples of hyperbolic but nonsystolic groups by showing that the triangle groups $(2,4,5)$ and $(2,5,5)$ are not systolic. Along the way we prove some results about subsets of systolic complexes stable under…

Given a class of compact spaces, we ask which groups can be maximal parabolic subgroups of a relatively hyperbolic group whose boundary is in the class. We investigate the class of 1-dimensional connected boundaries. We get that any…

Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably coHopfian. In particular, we show that the fundamental group of every…

We prove that if a countable group is elementarily equivalent to a non-abelian free group and all of its abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).

We identify a condition that prevents a hyperbolic space from being quasi-isometric to the curve complex of any non-sporadic surface. Our result applies to several hyperbolic complexes, including arc complexes, disk complexes,…

We refine Feighn--Handel's results on subgroups of mapping tori of free groups to the special case of free-by-cyclic groups. We use these refinements to show that any finitely generated free-by-cyclic group embeds in a {finitely generated…

For a group G relatively hyperbolic to a family of residually finite groups satisfying the Farrell-Jones conjecture, we reduce the solution of the Farrell-Jones conjecture for G to the case of certain nice cyclic extensions in G.

We construct the first known infinite family of quasi-isometry classes of subgroups of hyperbolic groups which are not hyperbolic and are of type $\mathrm{FP}(\mathbb{Q})$. We give a simple criterion for producing many non-hyperbolic…

Despite the significance of the notion of parabolic closures in Coxeter groups of finite ranks, the parabolic closure is not guaranteed to exist as a parabolic subgroup in a general case. In this paper, first we give a concrete example to…

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 exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear,…

In this paper we provide a procedure to obtain a non-trivial HHS structure on a hyperbolic space. In particular, we prove that given a finite collection $\mathcal{F}$ of quasi-convex subgroups of a hyperbolic group $G$, there is an HHG…

We give an example of a definable set in every free or torsion-free (non-elementary) hyperbolic group that is not in the Boolean algebra of equational sets. Hence, the theories of free and torsion-free (non-elementary) hyperbolic groups are…

We characterize relatively hyperbolic groups whose reduced $C^*$-algebra is simple as those, which have no non-trivial finite normal subgroups.

We show that if $H$ is a quasiconvex subgroup of infinite index in a non-elementary hyperbolic group $G$ then the Schreier coset graph $X$ for $G$ relative to $H$ is non-amenable (that is, $X$ has positive Cheeger constant). We present some…

We show that low-density random quotients of cubulated hyperbolic groups are again cubulated (and hyperbolic). Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the…

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 prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling.

We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has…