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

TL;DR

This paper classifies the large-scale geometric properties of pure mapping class groups of infinite-type surfaces, showing that their CB properties depend solely on the surface’s end structure and punctures.

Contribution

It provides a complete classification of pure mapping class groups' CB properties without additional assumptions, contrasting previous results for the full mapping class groups.

Findings

PMap(Σ) is globally CB only for the Loch Ness monster surface.

PMap(Σ) is locally CB or CB generated if Σ has finitely many ends and is not a punctured Loch Ness monster.

The classification depends solely on the surface's end structure and punctures.

Abstract

The work of Mann and Rafi gives a classification surfaces $\Sigma$ when $\textrm{Map}(\Sigma)$ 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 $\textrm{PMap}(\Sigma)$ 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.