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

The work of Mann and Rafi gives a classification surfaces $\mathrm{\Sigma }$ when $\text{Map}(\mathrm{\Sigma })$ 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 $\text{PMap}(\mathrm{\Sigma })$ is globally CB if and only if $\mathrm{\Sigma }$ is the Loch Ness monster surface, and locally CB or CB generated if and only if $\mathrm{\Sigma }$ has finitely many ends and is not a Loch Ness monster surface with (nonzero) punctures.