Frattini and related subgroups of Mapping Class Groups

TL;DR

This paper investigates the structure of certain subgroups within the Mapping Class Group of surfaces, specifically computing the intersection of all maximal finite index subgroups, and confirms a conjecture about their nilpotency.

Contribution

The paper computes the subgroup $\Phi_f(G)$ for specific subgroups of the Mapping Class Group and confirms Ivanov's conjecture on their nilpotency.

Findings

$\Phi_f(G)$ is nilpotent for the studied subgroups

Provides explicit computations of $\Phi_f(G)$ in the context of Mapping Class Groups

Answers Ivanov's question affirmatively for certain subgroups

Abstract

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