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

2019/09/16 by Sudipta Kolay, Kolay, Sudipta
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1909.07360

openalex publication_date 2019/09/16 · openalex created_date 2019/09/19 · openalex updated_date 2026/07/28

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.

Citations

• A finite set of generators for the homeotopy group of a 2-manifold

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• On squares of Dehn twists about non-separating curves of a non-orientable closed surface
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• A Presentation of the Mapping Class Groups