Minimal Sets of Generators for Big Mapping Class Groups

TL;DR

This paper determines minimal topological generating sets for the mapping class groups of various infinite-type surfaces, showing they can be generated by as few as two or three elements depending on the surface.

Contribution

It provides explicit minimal generating sets for the mapping class groups of infinite-genus surfaces with different numbers of ends, including new results for specific surfaces.

Findings

For n ≥ 8, mod(S(n)) is generated by three elements.

For n ≥ 3, mod(S(n)) is generated by four elements.

The Loch Ness Monster surface has a two-element generating set.

Abstract

Let $S(n)$ be the infinite-type surface with infinite genus and $n \in \mathbb{N}$ ends, all of which are accumulated by genus. The mapping class group of this surface, $\mod(S(n))$, is a Polish group that is not countably generated, but it is countably topologically generated. This paper focuses on finding minimal sets of generators for $\mod(S(n))$. We show that for $n \ge 8$, $\mod(S(n))$ is topologically generated by three elements, and for $n \ge 3$, $\mod(S(n))$ is topologically generated by four elements. We also establish a generating set of two elements for the Loch Ness Monster surface ($n=1$) and a generating set of three elements for the Jacob's Ladder surface ($n=2$).