Generating mapping class groups with elements of fixed finite order

TL;DR

This paper demonstrates that for large genus surfaces, the mapping class group can be generated by a small number of elements with fixed finite order, extending to other algebraic groups as well.

Contribution

It introduces new generation results for mapping class groups using elements of fixed finite order, including specific cases for order 5 and 6+ genus surfaces.

Findings

Mapping class groups of large genus surfaces are generated by three elements of order k (k ≥ 6).

Generation by four elements of order 5 is possible.

Similar generation results hold for permutation, linear, and automorphism groups.

Abstract

We show that for any $k$ at least $6$ and $g$ sufficiently large, the mapping class group of a surface of genus $g$ can be generated by three elements of order $k$. We also show that this can be done with four elements of order $5$. We additionally prove similar results for some permutation groups, linear groups, and automorphism groups of free groups.