Frattini and Related Subgroups of Mapping Class Groups

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