## Algebraic & Geometric Topology

### Algebraic and topological properties of big mapping class groups

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

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4109-4142.

Dates
Received: 1 December 2017
Revised: 23 April 2018
Accepted: 14 July 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1545102066

Digital Object Identifier
doi:10.2140/agt.2018.18.4109

Mathematical Reviews number (MathSciNet)
MR3892241

Zentralblatt MATH identifier
07006387

#### Citation

Patel, Priyam; Vlamis, Nicholas G. Algebraic and topological properties of big mapping class groups. Algebr. Geom. Topol. 18 (2018), no. 7, 4109--4142. doi:10.2140/agt.2018.18.4109. https://projecteuclid.org/euclid.agt/1545102066

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