Small Sets of Topological Generators for Big Mapping Class Groups

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, $\mathrm{M}\mathrm{a}\mathrm{p}(S(n))$, 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 $\mathrm{M}\mathrm{a}\mathrm{p}(S(n))$. We show that for $n\ge 8$, $\mathrm{M}\mathrm{a}\mathrm{p}(S(n))$ is topologically generated by three elements, and for $n\ge 3$, it is topologically generated by four elements. We also establish a generating set of two elements for the Loch Ness Monster surface $S(1)$, and a generating set of three elements for the Jacob’s Ladder surface $S(2)$.