Graphical models for topological groups: A case study on countable Stone spaces

TL;DR

This paper introduces Cayley--Abels--Rosendal graphs for Polish groups, establishing their properties and showing how they serve as topological analogues of finitely generated groups, with applications to homeomorphism groups of countable Stone spaces.

Contribution

It extends the concept of Cayley graphs to topological groups, providing a new geometric framework and characterizations for homeomorphism groups of countable Stone spaces.

Findings

Groups with Cayley--Abels--Rosendal graphs are quasi-isometric to their word metrics.

All considered homeomorphism groups are locally bounded.

Characterization of when these groups admit Cayley--Abels--Rosendal graphs.

Abstract

By analogy with the Cayley graph of a group with respect to a finite generating set or the Cayley--Abels graph of a totally disconnected, locally compact group, we detail countable connected graphs associated to Polish groups that we term Cayley--Abels--Rosendal graphs. A group admitting a Cayley--Abels--Rosendal graph acts on it continuously, coarsely metrically properly and cocompactly by isometries of the path metric. By an expansion of the Milnor--Schwarz lemma, it follows that the group is generated by a coarsely bounded set and the group equipped with a word metric with respect to a coarsely bounded generating set and the graph are quasi-isometric. In other words, groups admitting Cayley--Abels--Rosendal graphs are topological analogues of finitely generated groups. Our goal is to introduce this topological perspective on the work of Rosendal to a geometric group theorist. We apply these concepts to homeomorphism groups of countable Stone spaces. We completely characterize when these homeomorphism groups are coarsely bounded, when they are locally bounded (all of them are), and when they admit a Cayley--Abels--Rosendal graph, and if so produce a coarsely bounded generating set.