Frattini and related subgroups of mapping class groups

Let Γg,b denote the orientation-preserving mapping class group of a closed orientable surface of genus g with b punctures. For a group G let Φf(G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φf(G) is nilpotent when G is a finitely generated subgroup of Γg,b, in this paper we compute Φf(G) for certain subgroups of Γg,b. In particular, we answer Ivanov’s question in the affirmative for these subgroups of Γg,b.