The three boundedness classes of homeomorphism groups of countable Stone spaces are exactly the coarse equivalence classes, with the middle class quasi-isometric to the Hamming cube and infinite Hamming graphs bi-Lipschitz equivalent.
Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire , volume=
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A dense subset of the Gromov boundary of the grand arc graph is identified with geodesic laminations; the graph satisfies the bounded geodesic image theorem and its boundary is non-compact.
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Coarse geometry of homeomorphism groups: Classifying countable Stone spaces
The three boundedness classes of homeomorphism groups of countable Stone spaces are exactly the coarse equivalence classes, with the middle class quasi-isometric to the Hamming cube and infinite Hamming graphs bi-Lipschitz equivalent.