Algebraic and topological properties of big mapping class groups
By Priyam Patel, Nicholas G. Vlamis. Algebraic and Geometric Topology, 2017
59 citations recorded by the source corpus.
Let $S$ be an orientable, connected surface 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 $\mathrm{PMap}(S)$, detects the homeomorphism type of $S$. As a corollary, every automorphism of $\mathrm{PMap}(S)$ 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 $\mathrm{PMap}(S)$ is residually finite if and only if $S$ has finite genus, demonstrating that the algebraic structure of $\mathrm{PMap}(S)$ can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that $\mathrm{Map}(S)$ fails to be residually finite for any infinite-type surface $S$. In addition, we give a topological generating set for $\mathrm{PMap}(S)$ equipped with the compact-open topology. In particular, if $S$ has at most one end accumulated by genus, then $\mathrm{PMap}(S)$ is topologically generated by Dehn twists, otherwise the Dehn twists along with handle shifts topologically generate.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/bc1652f2498e62f4cb2885cacf2cfbb75e2ec2f0
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In Geometry and Topology
- Javier Aramayona
- Nicholas G. Vlamis
- Priyam Patel
- Kasra Rafi
- Kathryn Mann
- Samuel J. Taylor
- U. Hamenstaedt
- Saul Schleimer
In Mathematical Physics
Citations in the corpus
- The First Integral Cohomology of Pure Mapping Class Groups (2017, background): “The abelian group in the decomposition of PMap(S) presented in Corollary 6 is generated by handle shifts, which were introduced in [22].”
- Large-scale geometry of big mapping class groups (2019, background): “†/ consisting of mapping classes supported on finite-type subsurfaces of †; see [14].”