On the large scale geometry of big mapping class groups of surfaces with a unique maximal end

TL;DR

This paper characterizes the large scale geometry of big mapping class groups of surfaces with a unique maximal end, identifying conditions for them to be CB and providing explicit criteria and examples.

Contribution

It offers a complete characterization of globally CB big mapping class groups without the tameness condition and introduces criteria for CB generation.

Findings

Globally CB big mapping class groups are characterized for surfaces with a unique maximal end.

Locally CB big mapping class groups are CB generated for such surfaces.

An example of a non-tame surface with CB generated but not globally CB mapping class group.

Abstract

Building on the work of K. Mann and K. Rafi, we analyze the large scale geometry of big mapping class groups of surfaces with a unique maximal end. We obtain a complete characterization of those that are globally CB, which does not require the tameness condition. We prove that, for surfaces with a unique maximal end, any locally CB big mapping class group is CB generated, and we give an explicit criterion for determining which big mapping class groups are CB generated. Finally, we give an example of a non-tame surface whose mapping class group is CB generated but is not globally CB.