Roots of Dehn twists

TL;DR

This paper establishes elementary number-theoretic conditions for the existence of nth roots of Dehn twists in surface mapping class groups, revealing constraints on n and genus, and classifying roots for large n.

Contribution

It provides new number-theoretic criteria for roots of Dehn twists, extending previous work and classifying roots based on genus and order.

Findings

n must be odd for roots to exist

Maximum n achieved by Margalit-Schleimer roots

Roots exist for all genera above a certain threshold for given odd n

Abstract

D. Margalit and S. Schleimer found examples of roots of the Dehn twist about a nonseparating curve in a closed orientable surface, that is, homeomorphisms whose nth power is isotopic to the Dehn twist. Our main theorem gives elementary number-theoretic conditions that describe the values of n for which an nth root exists, given the genus of the surface. Among its applications, we show that n must be odd, that the Margalit-Schleimer roots achieve the maximum value of n among the roots for a given genus, and that for a given odd n, nth roots exist for all genera greater than (n-2)(n-1)/2. We also describe all nth roots having n greater than or equal to the genus.