Finite index subgroups of mapping class groups

Abstract

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(2 g-1) up to conjugation, a unique subgroup of index 2 g-1(2g+1) up to conjugation, and the other proper subgroups of ℳg, n are of index greater than 2 g-1(2g+1). In particular, the minimum index for a proper subgroup of ℳg, n is 2g-1(2g-1). © 2013 London Mathematical Society.