Frattini subgroups of hyperbolic-like groups

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