Roots of Dehn twists on nonorientable surfaces

TL;DR

This paper investigates the existence and classification of roots of Dehn twists on nonorientable surfaces, providing arithmetic descriptions and explicit constructions for roots of various degrees.

Contribution

It offers a new arithmetic framework for understanding roots of Dehn twists in the nonorientable setting and constructs explicit roots for large genus surfaces.

Findings

Roots of Dehn twists exist for sufficiently large genus surfaces.

Explicit formulas for roots of specific degrees are provided.

Roots of maximal degree are characterized for nonorientable surfaces.

Abstract

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist t_c about a nonseparating circle c in the mapping class group M(N_g) of a nonorientable surface N_g of genus g. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer n>1, then for each sufficiently large g, t_c has a root of degree n in M(N_g). Moreover, for any possible degree n we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in N_g.