Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 18, Number 7 (2018), 4109-4142.
Algebraic and topological properties of big mapping class groups
Let 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 is finite and at least , then the isomorphism type of the pure mapping class group associated to , denoted by , detects the homeomorphism type of . As a corollary, every automorphism of 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 is residually finite if and only if has finite genus, demonstrating that the algebraic structure of can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that fails to be residually finite for any infinite-type surface . In addition, we give a topological generating set for equipped with the compact-open topology. In particular, if has at most one end accumulated by genus, then is topologically generated by Dehn twists, otherwise it is topologically generated by Dehn twists along with handle shifts.
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4109-4142.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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