Related papers: Growth of relatively hyperbolic groups

Let $G$ be a toral relatively hyperbolic group, and let $\varphi\in\mathrm{Aut}(G)$. We prove that, under iteration of $\varphi$, the conjugacy length $||\varphi^n(g)||$ of every element $g\in G$ grows like $n^d\lambda^n$ for some…

We show that any partial ascending HNN extension of a free group embeds in an actual ascending HNN extension of a free group. Moreover, we can ensure that it embeds as the parabolic subgroup of a relatively hyperbolic group.

We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain…

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.

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.

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\mathbb{P}=\{P_1,\dots,P_m\}$. Let $H_1,H_2$ be subgroups of $G$ such that $H_1$ is relatively quasiconvex with respect to $\mathbb{P}$ and…

We introduce a coarse flow space for relatively hyperbolic groups and use it to verify a regularity condition for the action of relatively hyperbolic groups on their boundaries. As an application the Farrell-Jones Conjecture for relatively…

This paper overviews recent developments in the classification up to quasi-isometry of finitely generated groups, and more specifically of relatively hyperbolic groups.

We prove that the fundamental group of a finite graph of convergence groups with parabolic edge groups is a convergence group. Using this result, under some mild assumptions, we prove a combination theorem for a graph of convergence groups…

We describe an algorithm which determines whether or not a group which is hyperbolic relative to abelian groups admits a nontrivial splitting over a finite group.

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…

A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis…

We study the set of homomorphisms from a fixed finitely generated group into a family of groups which are `uniformly acylindrically hyperbolic'. Our main results reduce this study to sets of homomorphisms which do not diverge in an…

We prove that a finitely generated solvable group which is not virtually nilpotent has exponential conjugacy growth.

This note records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely…

In this paper, we consider the conjugacy growth function of a group, which counts the number of conjugacy classes which intersect a ball of radius $n$ centered at the identity. We prove that in the case of virtually polycyclic groups, this…

We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups.

We prove that for a relatively hyperbolic group G there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of G. Under natural assumptions, the same conclusion holds for the…

Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and $gr(H,A,B)$ be the growth function of the double cosets $AhB, h \in H$. We prove that the behavior of $gr(H,A,B)$ splits into two different cases. If $A$ and $B$ are not…

We prove that the semistability growth of hyperbolic groups is linear, which implies that hyperbolic groups which are sci (simply connected at infinity) have linear sci growth. Based on the linearity of the end-depth of finitely presented…