Subgroups of the mapping class group of the torus generated by powers of Dehn twists

TL;DR

This paper investigates subgroups of the torus's mapping class group generated by powers of Dehn twists, providing criteria for freeness, characterizations of specific subgroups, and classifications based on intersection numbers.

Contribution

It introduces a criterion for when powers of Dehn twists generate free groups and characterizes subgroups generated by specific powers based on intersection properties.

Findings

Subgroups generated by three uniform powers can be the entire group, a direct product, or a free group of rank at most three.

Criteria using the ping pong lemma determine when powers generate free groups.

Characterizations of subgroups generated by squares or other powers based on intersection numbers.

Abstract

We study subgroups of the mapping class group of the torus generated by powers generated by powers of Dehn twists. We give a criterion to show when a collection of powers Dehn twists generates a free group using the ping pong lemma. We show that the subgroup generated by three uniform powers of Dehn twists can be either the whole mapping class group, a direct product of a free group of rank two with the cyclic group of order two, or a free group of rank at most three. We characterize subgroups generated by a collection of (respectively squares of) Dehn twists, if there is a pair among so that the geometric intersection number of the corresponding simple closed curves is two (respectively one). We also determine the subgroup generated by uniform powers of all Dehn twists.