On roots of Dehn twists

TL;DR

This paper characterizes the conjugacy classes of roots of Dehn twists about nonseparating curves, linking them to periodic maps, and provides data to determine their class, including degrees and explicit examples.

Contribution

It establishes a correspondence between roots of Dehn twists and periodic maps, and provides a method to classify and explicitly construct roots, including degree constraints.

Findings

Conjugacy classes of roots correspond to classes of periodic maps.

The degree of roots must be odd.

Explicit Dehn twist expression for roots of degree 3.

Abstract

Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Futhermore, we give data set which determine the conjugacy class of a root. As a consequence, we can find the minimum degree and the maximum degree, and show that the degree must be odd. Also, we give Dehn twist expression of the root of degree 3.