Topological normal generation of big mapping class groups

Abstract

A topological group $G$ is topologically normally generated if there exists $g \in G$ such that the normal closure of $g$ is dense in $G$. Let $S$ be a tame, infinite type surface whose mapping class group $\mathrm{Map}(S)$ is generated by a coarsely bounded set (CB generated). We prove that if the end space of $S$ is countable, then $\mathrm{Map}(S)$ is topologically normally generated if and only if $S$ is uniquely self-similar. Moreover, when the end space of $S$ is uncountable, we provide a sufficient condition under which $\mathrm{Map}(S)$ is topologically normally generated. As a consequence, we construct uncountably many examples of surfaces that are not telescoping yet have topologically normally generated mapping class groups. Additionally, we establish the semidirect product structure of $\mathrm{FMap}(S)$, the subgroup of $\mathrm{Map}(S)$ that pointwisely fixes all maximal ends that each is isolated in the set of maximal ends of $S$. This result leads to a proof that the minimum number of topological normal generators of $\mathrm{Map}(S)$ is bounded both above and below by constants that depend only on the topology of $S$. Furthermore, we demonstrate that the upper bound grows quadratically with respect to this constant.