Torsion Elements in the Mapping Class Group of a Surface

TL;DR

This paper investigates the torsion elements in the mapping class group of a surface with marked points, establishing conditions under which these elements generate the entire group, with a specific exception.

Contribution

It characterizes when torsion elements generate the mapping class group of a surface with marked points, identifying a unique exception case.

Findings

Torsion elements generate the mapping class group except when (g, r) = (2, 5k+4).

Provides a complete characterization of generating sets involving torsion elements.

Identifies a specific exceptional case in the structure of the mapping class group.

Abstract

Given a finite set of $r$ points in a closed surface of genus $g$, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class group if and only if $(g, r) \neq (2, 5k+4)$ for some integer $k$.