Finite index subgroups of mapping class groups

TL;DR

This paper classifies the finite index subgroups of the mapping class group of surfaces with genus g ≥ 3, identifying unique minimal index subgroups and establishing lower bounds for other proper subgroups.

Contribution

It determines the exact indices and conjugacy classes of the smallest proper subgroups of the mapping class group for genus g ≥ 3.

Findings

Unique subgroups of specific indices up to conjugation

Minimal index for proper subgroups is 2^{g-1}(2^{g}-1)

Other proper subgroups have index greater than 2^{g-1}(2^{g}+1)

Abstract

Let $g\geq3$ and $n\geq0$, and let ${\mathcal{M}}_{g,n}$ be the mapping class group of a surface of genus $g$ with $n$ boundary components. We prove that ${\mathcal{M}}_{g,n}$ contains a unique subgroup of index $2^{g-1}(2^{g}-1)$ up to conjugation, a unique subgroup of index $2^{g-1}(2^{g}+1)$ up to conjugation, and the other proper subgroups of ${\mathcal{M}}_{g,n}$ are of index greater than $2^{g-1}(2^{g}+1)$. In particular, the minimum index for a proper subgroup of ${\mathcal{M}}_{g,n}$ is $2^{g-1}(2^{g}-1)$.