Finite index subgroups of mapping class groups

Let g ≥ 3 and n ≥ 0, and let Mg,n be the mapping class group of a surface of genus g with n boundary components. We prove that Mg,n contains a unique subgroup of index 2g−1(2g − 1) up to conjugation, a unique subgroup of index 2g−1(2g + 1) up to conjugation, and the other proper subgroups ofMg,n are of index greater than 2g−1(2g+1). In particular, the minimum index for a proper subgroup of Mg,n is 2g−1(2g − 1). AMS Subject Classification. Primary: 57M99. Secondary: 20G40, 20E28. 0 Introduction and statement of results The interaction between mapping class groups and finite groups has long been a topic of interest. The famous Hurwitz bound of 1893 showed that the mapping class group of 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 [19], [20]. The subject of finite index subgroups of mapping class groups was brought into focus by Grossman’s discovery that the mapping class group Mg,n = π0(Homeo(Σg,n)) of an oriented surface Σg,n of genus g and n boundary components is residually finite, and thus well-endowed with subgroups of finite index [15]. (Homeo(Σg,n) denotes the space of those homeomorphisms of Σg,n that preserve the orientation and are the identity on the boundary.) This prompts the “dual” question: What is the minimum index mi(Mg,n) of a proper subgroup of finite index in Mg,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 is mi(Mg,n) > 4g + 4 for g ≥ 3 (see [26]). This inequality is used by Aramayona and Souto to prove that, if g ≥ 6 and g′ ≤ 2g − 1, then any nontrivial homomorphism Mg,n →Mg′,n′ is induced by an embedding [1]. It is also an important ingredient in the proof of Zimmermann [33] that, for g = 3 and 4, the minimal nontrivial quotient of Mg,0 is Sp2g(F2). The “headline” result of this paper is the following exact, exponential bound. Theorem 0.1. For g ≥ 3 and n ≥ 0, mi(Mg,n) = mi(Sp2g(Z)) = mi(Sp2g(F2)) = 2g−1(2g − 1) . This exponential bound is all the more surprising since in similar questions we get linear (expected) bounds. For instance, Bridson [5, 6] has proved that a mapping class group of a surface