#### Volume 18, issue 7 (2018)

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Algebraic and topological properties of big mapping class groups

### Priyam Patel and Nicholas G Vlamis

Algebraic & Geometric Topology 18 (2018) 4109–4142
##### Abstract

Let $S$ be an orientable, connected topological surface of infinite type (that is, 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 $S\phantom{\rule{0.3em}{0ex}}$, denoted by $PMap\left(S\right)$, detects the homeomorphism type of $S\phantom{\rule{0.3em}{0ex}}$. As a corollary, every automorphism of $PMap\left(S\right)$ 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 $PMap\left(S\right)$ is residually finite if and only if $S$ has finite genus, demonstrating that the algebraic structure of $PMap\left(S\right)$ can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that $Map\left(S\right)$ fails to be residually finite for any infinite-type surface $S\phantom{\rule{0.3em}{0ex}}$. In addition, we give a topological generating set for $PMap\left(S\right)$ equipped with the compact-open topology. In particular, if $S$ has at most one end accumulated by genus, then $PMap\left(S\right)$ is topologically generated by Dehn twists, otherwise it is topologically generated by Dehn twists along with handle shifts.

##### Keywords
mapping class groups, infinite-type surfaces, topological groups
##### Mathematical Subject Classification 2010
Primary: 20E26, 37E30, 57M07, 57S05