Roots of crosscap slides and crosscap transpositions

TL;DR

This paper investigates the roots of crosscap slides and transpositions within the mapping class group of nonorientable surfaces, providing conditions for their existence and deepening understanding of their algebraic structure.

Contribution

It establishes necessary and sufficient conditions for roots of crosscap slides and transpositions, expanding knowledge of the algebraic properties of mapping class groups.

Findings

Criteria for the existence of roots of crosscap slides.

Criteria for the existence of roots of crosscap transpositions.

Enhanced understanding of the algebraic structure of mapping class groups.

Abstract

Let $N_{g}$ denote a closed nonorientable surface of genus $g$. For $g \geq 2$ the mapping class group $\mathcal{M}(N_{g})$ is generated by Dehn twists and one crosscap slide ($Y$-homeomorphism) or by Dehn twists and a crosscap transposition. Margalit and Schleimer observed that Dehn twists have nontrivial roots. We give necessary and sufficient conditions for the existence of a root of a crosscap slide and a crosscap transposition.