Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

Abstract

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 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 locally coarsely bounded pure mapping class groups of infinite rank graphs. This dimension is always either zero or infinite.

arXiv comment51 pages, 16 figures. v4: new proof of Lemma 2.3 and other minor edits. To appear in Advances in Mathematics

References in corpus (6)

• Metric geometry of locally compact groups
• Large scale geometry of big mapping class groups
• The first integral cohomology of pure mapping class groups
• Algebraic and topological properties of big mapping class groups
• Graphs of curves on infinite-type surfaces with mapping class group actions
• The Automorphism Group of the Infinite-Rank Free Group is Coarsely Bounded

Cited by in corpus (2)

• Graphical models for topological groups: A case study on countable Stone spaces
• Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs