Small Torsion Topological Generators for Big Mapping Class Groups

TL;DR

This paper investigates minimal topological generating sets of infinite-type surface mapping class groups using torsion elements, establishing bounds on the number and order of generators needed for various surfaces.

Contribution

It proves that certain infinite-type surface mapping class groups are generated by a small number of torsion elements, including involutions, with explicit bounds depending on the surface.

Findings

Map(S(n)) is topologically generated by four involutions for all n ≥ 16.

Map(S(n)) is generated by three involutions for n=1 and n=2.

For even n ≥ 8, generated by four torsion elements of order n.

Abstract

Let $S(n)$, for $n \in \mathbb{N}$, be the infinite-type surface of infinite genus with $n$ ends, each accumulated by genus. Although the mapping class groups of these surfaces are not countably generated,they are Polish groups and hence admit a countable topological generating set. We study minimal topological generating sets for $\mathrm{Map}(S(n))$ consisting entirely of torsion elements, with special attention to involutions. In particular, we prove that $\mathrm{Map}(S(n))$ is topologically generated by four involutions for all $n \geq 16$, and by three involutions for the Loch Ness Monster surface ($n = 1$) and the Jacob's Ladder surface ($n = 2$). We also establish that for even $n \geq 8$, $\mathrm{Map}(S(n))$ is topologically generated by four torsion elements of order $n$. For odd $n \geq 8$, it is topologically generated by three torsion elements of order $n$ and one torsion element of order $n - 1$.