Topological normal generation of big mapping class groups

TL;DR

This paper investigates the conditions under which big mapping class groups are topologically normally generated, revealing a connection with the end space structure of the surface and providing bounds on the number of generators.

Contribution

It characterizes topological normal generation of big mapping class groups based on surface end space properties and establishes bounds on the number of generators needed.

Findings

Topological normal generation linked to countability of end space and self-similarity.

Constructs examples of non-telescoping surfaces with normally generated groups.

Bounds on the number of generators depend quadratically on surface topology.

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.