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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

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.

mapping class groups, infinite-type surfaces, topological groups
Mathematical Subject Classification 2010
Primary: 20E26, 37E30, 57M07, 57S05
Received: 1 December 2017
Revised: 23 April 2018
Accepted: 14 July 2018
Published: 11 December 2018
Priyam Patel
Department of Mathematics
University of California, Santa Barbara
Santa Barbara, CA
United States
Nicholas G Vlamis
Department of Mathematics
Queens College, CUNY
Flushing, NY
United States