The paper defines Cayley-Abels-Rosendal graphs for Polish groups and uses them to classify when homeomorphism groups of countable Stone spaces are coarsely bounded, locally bounded, and boundedly generated.
Metric geometry of locally compact groups , volume 25 of EMS Tracts in Mathematics
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Graphical models for topological groups: A case study on countable Stone spaces
The paper defines Cayley-Abels-Rosendal graphs for Polish groups and uses them to classify when homeomorphism groups of countable Stone spaces are coarsely bounded, locally bounded, and boundedly generated.