Frattini subgroups of hyperbolic-like groups

Abstract

We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group $G$ admitting a general type action on a hyperbolic space $S$, we show that the induced action of the Frattini subgroup $\hat{I}¦(G)$ on $S$ has bounded orbits. This implies that $\hat{I}¦(G)$ is "small" compared to $G$; in particular, $|G:\hat{I}¦(G)|=\infty$. In contrast, for any finitely generated non-cyclic group $Q$ with $\hat{I}¦(Q)=\{1\}$, we construct an infinite lacunary hyperbolic group $L$ such that $L/\hat{I}¦(L)\cong Q$; in particular, the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups.

Final version, to appear in the Journal of Algebra