Small index subgroups of the mapping class group

TL;DR

This paper proves that for closed oriented surfaces of genus at least 3, the mapping class group does not have any proper subgroups with an index less than or equal to four times the genus plus four.

Contribution

It establishes a new lower bound on the index of proper subgroups in the mapping class group for genus at least 3.

Findings

No proper subgroup of index ≤ 4ρ+4 exists for genus ρ ≥ 3

Provides bounds on subgroup indices in the mapping class group

Advances understanding of the subgroup structure of the mapping class group

Abstract

We prove that the mapping class group of a closed oriented surface of genus $\rho \ge 3$ has no proper subgroup of index $\le 4 \rho +4$.