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 $\Phi(G)$ on $S$ has bounded orbits. This implies that $\Phi(G)$ is "small" compared to $G$; in particular, $|G:\Phi(G)|=\infty$. In contrast, for any finitely generated non-cyclic group $Q$ with $\Phi(Q)={ 1}$, we construct an infinite lacunary hyperbolic group $L$ such that $L/\Phi(L)\cong Q$; 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.