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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.
Keywords: mapping class groups, infinite-type surfaces, topological groups
Patel, Priyam 1 ; Vlamis, Nicholas 2
@article{10_2140_agt_2018_18_4109,
author = {Patel, Priyam and Vlamis, Nicholas},
title = {Algebraic and topological properties of big mapping class groups},
journal = {Algebraic and Geometric Topology},
pages = {4109--4142},
publisher = {mathdoc},
volume = {18},
number = {7},
year = {2018},
doi = {10.2140/agt.2018.18.4109},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.4109/}
}
TY - JOUR AU - Patel, Priyam AU - Vlamis, Nicholas TI - Algebraic and topological properties of big mapping class groups JO - Algebraic and Geometric Topology PY - 2018 SP - 4109 EP - 4142 VL - 18 IS - 7 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.4109/ DO - 10.2140/agt.2018.18.4109 ID - 10_2140_agt_2018_18_4109 ER -
%0 Journal Article %A Patel, Priyam %A Vlamis, Nicholas %T Algebraic and topological properties of big mapping class groups %J Algebraic and Geometric Topology %D 2018 %P 4109-4142 %V 18 %N 7 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.4109/ %R 10.2140/agt.2018.18.4109 %F 10_2140_agt_2018_18_4109
Patel, Priyam; Vlamis, Nicholas. Algebraic and topological properties of big mapping class groups. Algebraic and Geometric Topology, Tome 18 (2018) no. 7, pp. 4109-4142. doi: 10.2140/agt.2018.18.4109
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