Algebraic and topological properties of big mapping class groups

Abstract

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.

32 pages, 3 figures; v2 has several minor changes and corrections, including a more explicit treatment of the centers of big mapping class groups in Section 3

References in corpus (1)

• Gromov-Hyperbolicity of the ray graph and quasimorphisms on a big mapping class group

Cited by in corpus (13)

• Big mapping class groups: an overview
• Large scale geometry of big mapping class groups
• The first integral cohomology of pure mapping class groups
• Isometry groups of infinite-genus hyperbolic surfaces
• Big mapping class groups and rigidity of the simple circle
• Mapping class groups with the Rokhlin property
• Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs
• Large-Scale Geometry of Pure Mapping Class Groups of Infinite-Type Surfaces
• Mapping class groups of surfaces with noncompact boundary components
• Infinite-type loxodromic isometries of the relative arc graph
• Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs
• Non-planar ends are continuously unforgettable
• Multitwists in big mapping class groups