On the Large Scale Geometry of Big Mapping Class Groups of Surfaces with a Unique Maximal End

Building on the work of Mann and Rafi [MR23], 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 answer [MR23, Problem 6.12] by giving an example of a non-tame surface whose mapping class group is CB generated but is not globally CB.