Frattini and related subgroups of Mapping Class Groups

Abstract

Let $\hat{I}{“}_{g,b}$ denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus $g$ with $b$ punctures. For a group $G$ let $\hat{I}{¦}_f(G)$ denote the intersection of all maximal subgroups of finite index in $G$. Motivated by a question of Ivanov as to whether $\hat{I}{¦}_f(G)$ is nilpotent when $G$ is a finitely generated subgroup of $\hat{I}{“}_{g,b}$, in this paper we compute $\hat{I}{¦}_f(G)$ for certain subgroups of $\hat{I}{“}_{g,b}$. In particular, we answer Ivanov's question in the affirmative for these subgroups of $\hat{I}{“}_{g,b}$.

12 pages. v2: abstract and introduction slightly rewritten