Large-Scale Geometry of Pure Mapping Class Groups of Infinite-Type Surfaces

Abstract

The work of Mann and Rafi gives a classification surfaces $\Sigma$ when $\text{Map}(\Sigma )$ is globally CB, locally CB, and CB generated under the technical assumption of tameness. In this article, we restrict our study to the pure mapping class group and give a complete classification without additional assumptions. In stark contrast with the rich class of examples of Mann--Rafi, we prove that $\text{PMap}(\Sigma )$ is globally CB if and only if $\Sigma$ is the Loch Ness monster surface, and locally CB or CB generated if and only if $\Sigma$ has finitely many ends and is not a Loch Ness monster surface with (nonzero) punctures.

arXiv comment14 pages, 6 figures. Comments welcome!

References in corpus (9)

• Geometry of the Complex of Curves I: Hyperbolicity
• Metric geometry of locally compact groups
• Big mapping class groups: an overview
• Large scale geometry of big mapping class groups
• The first integral cohomology of pure mapping class groups
• Algebraic and topological properties of big mapping class groups
• Large-Scale Geometry of Pure Mapping Class Groups of Infinite-Type Surfaces
• Automatic continuity of pure mapping class groups
• On the large scale geometry of big mapping class groups of surfaces with a unique maximal end

Cited by in corpus (2)

• Large-Scale Geometry of Pure Mapping Class Groups of Infinite-Type Surfaces
• Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs