Speaker: Ekaterina Rybak
Title: Frattini subgroups of hyperbolic-like groups.
Abstract: The Frattini subgroup $\Phi(G)$ of a group G is the intersection of all maximal subgroups of G; if G has no maximal subgroups, $\Phi(G)=G$ by definition. Frattini subgroups of groups with ``hyperbolic-like" geometry are often small in a suitable sense. Generalizing several known results, we prove that for any countable group $G$ admitting a general type action on a hyperbolic space S, the induced action of the Frattini subgroup $\Phi(G)$ on $S$ has bounded orbits, in particular, $\Phi(G)$ has infinite index in G. In contrast, we show that 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. The talk is based on a joint work with Gil Goffer and Denis Osin.