Small index subgroups of the mapping class group

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...

A note on the abelianizations of finite-index subgroups of the mapping class group

Let Σ g,b be an oriented genus g surface with b boundary components and p punctures and let Mod(Σ g,b) be its mapping class group, that is, the group of isotopy classes of orientation–preserving diffeomorphisms of Σ g,b that fix the boundary components and punctures pointwise (we will omit b or p if they vanish). A long–standing conjecture of Ivanov (see [7] for a recent discussion) says that f...

Finite abelian subgroups of the mapping class group

Let S be a closed, smooth, orientable surface of genus 2. The mapping class group M of S (or MCG) is the group of isotopy classes of homeomorphisms of S . In this paper we shall investigate the conjugacy classes of finite elementary abelian subgroups, and more generally the finite abelian subgroups of M . While the general finite subgroup classification is important we focus on the case where G...

Abelian and Solvable Subgroups of the Mapping Class Group Joan

Curve Complexes and Finite Index Subgroups of Mapping Class Groups

Let Mod(S) be the extended mapping class group of a surface S. For S the twice-punctured torus, we show that there exists an isomorphism of finite index subgroups of Mod(S) which is not the restriction of any inner automorphism. For S a torus with at least three punctures, we show that every injection of a finite index subgroup of Mod(S) into Mod(S) is the restriction of an inner automorphism o...