# Algebraic and topological properties of big mapping class groups

## TL;DR

This paper investigates the algebraic and topological properties of big mapping class groups for infinite-type surfaces, showing how their structure can detect surface topology and distinguishing finite from infinite genus cases.

## Contribution

It proves that the pure mapping class group detects the surface's homeomorphism type for certain infinite-type surfaces and characterizes when these groups are residually finite.

## Key findings

- Pure mapping class group detects surface homeomorphism type for genus ≥ 4.
- Residually finiteness of PMap(S) characterizes finite genus surfaces.
- PMap(S) is not residually finite for any infinite-type surface.

## Abstract

Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$, denoted $\mathrm{PMap}(S)$, detects the homeomorphism type of $S$. As a corollary, every automorphism of $\mathrm{PMap}(S)$ is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that $\mathrm{PMap}(S)$ is residually finite if and only if $S$ has finite genus, demonstrating that the algebraic structure of $\mathrm{PMap}(S)$ can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that $\mathrm{Map}(S)$ fails to be residually finite for any infinite-type surface $S$. In addition, we give a topological generating set for $\mathrm{PMap}(S)$ equipped with the compact-open topology. In particular, if $S$ has at most one end accumulated by genus, then $\mathrm{PMap}(S)$ is topologically generated by Dehn twists, otherwise the Dehn twists along with handle shifts topologically generate.