Factoring periodic maps into Dehn twists

TL;DR

This paper introduces methods to factor periodic mapping classes into Dehn twists and explores their roots, revealing conjugates of certain periodic maps whose products are pseudo-Anosov, advancing understanding of surface mapping class groups.

Contribution

It develops new techniques for factoring periodic mapping classes into Dehn twists and analyzes roots of Dehn twists, including existence results for conjugates with pseudo-Anosov products.

Findings

Methods for factoring periodic classes into Dehn twists

Factoring roots of Dehn twists as Dehn twist words

Existence of conjugates of certain periodic maps with pseudo-Anosov products

Abstract

Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g \geq 1$. In this paper, we develop various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy. As applications, we develop methods for factoring certain roots of Dehn twists as words in Dehn twists. We will also show the existence of conjugates of periodic maps of order $4g$ and $4g+2$, for $g\geq 2$, whose product is pseudo-Anosov.