Large scale geometry of big mapping class groups

Abstract

We study the large-scale geometry of mapping class groups of surfaces of infinite type, using the framework of Rosendal for coarse geometry of non locally compact groups. We give a complete classification of those surfaces whose mapping class groups have local coarse boundedness (the analog of local compactness). When the end space of the surface is countable or tame, we also give a classification of those surface where there exists a coarsely bounded generating set (the analog of finite or compact generation, giving the group a well-defined quasi-isometry type) and those surfaces with mapping class groups of bounded diameter (the analog of compactness).

numerous small corrections in v2

References in corpus (9)

• Geometry of the Complex of Curves I: Hyperbolicity
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• Gromov-Hyperbolicity of the ray graph and quasimorphisms on a big mapping class group
• Algebraic and topological properties of big mapping class groups
• Graphs of curves on infinite-type surfaces with mapping class group actions
• Wild translation surfaces and infinite genus
• Large scale geometry of metrisable groups
• Big mapping class groups acting on homology

Cited by in corpus (18)

• Big mapping class groups: an overview
• Big mapping class groups with hyperbolic actions: classification and applications
• Loxodromic elements in big mapping class groups via the Hooper-Thurston-Veech construction
• Right-angled Artin groups as normal subgroups of mapping class groups
• 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
• Self-similar surfaces: involutions and perfection
• Curve graphs of surfaces with finite-invariance index 1
• Big mapping class groups with uncountable integral homology
• Three perfect mapping class groups
• On the homology of big mapping class groups
• Graphical models for topological groups: A case study on countable Stone spaces
• Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs
• A new construction of subgroups of big mapping class groups
• Twisted Rokhlin property for mapping class groups
• Asymptotic Dimension of Big Mapping Class Groups
• A class of open surfaces with algorithmically solvable homeomorphism problem