Finite index subgroups of mapping class groups

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 \(2^{g-1}(2^{g}-1)\).