Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

We discuss the large-scale geometry of pure mapping class groups of locally finite, infinite graphs, motivated by recent work of Algom-Kfir--Bestvina and the work of Mann--Rafi on the large-scale geometry of mapping class groups of infinite-type surfaces. Using the framework of Rosendal for coarse geometry of non-locally compact groups, we classify when the pure mapping class group of a locally finite, infinite graph is globally coarsely bounded (an analog of compact) and when it is locally coarsely bounded (an analog of locally compact). Our techniques also give lower bounds on the first integral cohomology of the pure mapping class group for some graphs and allow us to compute the asymptotic dimension of all of the locally coarsely bounded pure mapping class groups of infinite rank graphs. This dimension is always either zero or infinite.Comment: 50 pages, 14 figures. Added a new section 9 computing asymptotic dimension for locally CB pure mapping class groups. Also made several edits throughout the document and included new figure