Large-scale geometry of pure mapping class groups of infinite-type surfaces

Abstract:

The work of Mann and Rafi [Geom. Topol. 27 (2023), pp. 2237–2296] gives a classification of surfaces $\Sigma$ when $\mathrm {Map}(\Sigma )$ 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 $\mathrm {PMap}(\Sigma )$ 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.