Finite index subgroups of mapping class groups

Abstract

Let $g\ge 3$ and $n\ge 0$, 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 $2^{g-1}(2^g-1)$.

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