# Finite index subgroups of mapping class groups

Volker Gebhardt
University of Western Sydney
3 April 2012, 12 noon - 1pm, Room 2.39, Building Y, Kingswood Campus, University of Western Sydney

## Abstract

The interaction between mapping class groups and finite groups has long been a topic of interest. The famous Hurwitz bound of 1893 states that a closed Riemann surface of genus $$g$$ has an upper bound of $$84(g-1)$$ for the order of its finite subgroups, and Kerckhoff showed that the order of finite cyclic subgroups is bounded above by $$4g+2$$.

The subject of finite index subgroups of mapping class groups was brought into focus by Grossman's discovery that the mapping class group $${\mathcal{M}}_{g,n}$$ of an oriented surface $$\Sigma_{g,n}$$ of genus $$g$$ with $$n$$ boundary components is residually finite, and thus well-endowed with subgroups of finite index. This prompts the "dual" question: What is the minimum index $$\mathrm{mi}({\mathcal{M}}_{g,n})$$ of a proper subgroup of finite index in $${\mathcal{M}}_{g,n}$$?

Results to date have suggested that, like the maximum finite order question, the minimum index question should have an answer that is linear in $$g$$. The best previously published bound due to Paris is $$\mathrm{mi}({\mathcal{M}}% _{g,n})>4g+4$$ for $$g\geq 3$$. This inequality is used by Aramayona and Souto to prove that, if $$g\geq 6$$ and $$g^{\prime}\leq 2g-1$$, then any nontrivial homomorphism $${\mathcal{M}}_{g,n}\rightarrow{\mathcal{M}}_{g^{\prime},n^{\prime}}$$ is induced by an embedding. It is also an important ingredient in the proof of Zimmermann that, for $$g=3$$ and $$4$$, the minimal nontrivial quotient of $${\mathcal{M}}_{g,0}$$ is $$\mathrm{Sp}_{2g}({\mathbb{F}}_{2})$$.

I will report on recent work with Jon Berrick and Luis Paris, in which we showed an exact exponential bound for $$\mathrm{mi}({\mathcal{M}}_{g,n})$$. Specifically, we proved 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 $$2^{g-1}(2^{g}-1)$$.