Algebraic & Geometric Topology

Algebraic and topological properties of big mapping class groups

Priyam Patel and Nicholas G Vlamis

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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

Subjects
Primary: 20E26: Residual properties and generalizations; residually finite groups 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 57M07: Topological methods in group theory 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms

Keywords
mapping class groups infinite-type surfaces topological groups

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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