Actions of maximal growth of hyperbolic groups

TL;DR

This paper demonstrates that non-elementary hyperbolic groups can act with maximal growth on certain sets with finite orbits, and explores conditions for quasiconvex subgroups to generate free products with infinite index.

Contribution

It establishes the existence of maximal growth actions for hyperbolic groups and characterizes quasiconvex subgroups forming free products under specific conditions.

Findings

Hyperbolic groups act with maximal growth on sets with finite orbits.

Existence of elements generating quasiconvex free products with infinite index.

Characterization of when quasiconvex subgroups form free products with infinite index.

Abstract

We prove that every non-elementary hyperbolic group $G$ acts with maximal growth on some set $X$ such that every orbit of any element $g \in G$ is finite. As a side-product of our approach we prove that if $G$ is non-elementary hyperbolic, $\HH \leq G$ is quasiconvex of infinite index then there exists $g \in G$ such that $<\HH,g>$ is quasiconvex of infinite index and is isomorphic to $\HH*<g >$ if and only if $\HH \cap E(G)= \{e\} $, where $E(G)$ is the maximal finite normal subgroup of $G$.