Actions of maximal growth of hyperbolic groups

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* $ if and only if $\HH \cap E(G)= \{e\} $, where $E(G)$ is the maximal finite normal subgroup of $G$.