All finite groups are involved in the Mapping Class Group

TL;DR

This paper proves that for each fixed genus, the mapping class group contains finite index subgroups whose quotients include every finite group, revealing the group's vast algebraic complexity.

Contribution

It establishes that all finite groups are involved in the mapping class group for fixed genus, a significant advancement in understanding its subgroup structure.

Findings

Every finite group occurs as a quotient of a finite index subgroup of the mapping class group.

The result holds for each fixed genus g ≥ 1.

Mapping class groups are algebraically rich, encompassing all finite groups as quotients.

Abstract

Let $\Gamma_g$ denote the orientation-preserving Mapping Class Group of the genus $g\geq 1$ closed orientable surface. In this paper we show that for fixed $g$, every finite group occurs as a quotient of a finite index subgroup of $\Gamma_g$.