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.
On a lemma of Milnor and Schwarz, apr \`e s Rosendal
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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.