On the commutator length of a Dehn twist

TL;DR

This paper investigates the algebraic properties of Dehn twists in the mapping class groups of surfaces, showing that certain powers of these twists can be expressed as single commutators, revealing new structural insights.

Contribution

It establishes that on nonorientable surfaces of genus at least 7, powers of Dehn twists are single commutators, extending to orientable surfaces under specific conditions.

Findings

Powers of Dehn twists are single commutators in certain nonorientable surfaces.

Results apply to the twist subgroup and extended mapping class group.

The findings hold for surfaces with genus at least 7 (nonorientable) and 3 (orientable).

Abstract

We show that on a nonorientable surface of genus at least 7 any power of a Dehn twist is equal to a single commutator in the mapping class group and the same is true, under additional assumptions, for the twist subgroup, and also for the extended mapping class group of an orientable surface of genus at least 3.