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 $Γ_{g,b}$.

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