This paper classifies the large-scale geometry of symmetry groups of countable Stone spaces. A Stone space is a compact topological space made from countable discrete pieces, and each such space is described by two simple pieces of data: a Cantor–Bendixson rank α (a countable ordinal that measures how points cluster) and an integer n≥1 that counts the points of maximal rank. The authors show that the homeomorphism groups of these spaces (the groups of continuous self-maps) fall into exactly three coarse equivalence classes determined by α and n.

Concretely, the three cases are: when n=1 the group is coarsely bounded and so is quasi-isometric to a single point; when n>1 and α is a successor ordinal the group is quasi-isometric to the countably infinite Hamming cube; and when n>1 and α is a limit ordinal the group is coarsely equivalent to a certain “one-ended tree of Hamming cubes.” The countably infinite Hamming cube is the graph whose vertices are binary sequences with only finitely many 1s, with edges joining sequences that differ in a single coordinate. The paper also proves that infinite Hamming graphs built from any finite alphabet are essentially the same shape (bi-Lipschitz equivalent).

To reach these conclusions the authors build explicit geometric models for the groups. They use versions of Cayley graphs adapted to topological groups, called Cayley–Abels–Rosendal graphs, and for groups that are not generated by coarsely bounded sets they use a coarse variant introduced by Kopreski and Shaji. One guiding example is the two-point compactification of the integers. In that example vertices of the model graph correspond to ways of splitting the space into two clopen pieces, and those splits differ only on finitely many points. Recording those finitely many differences identifies the graph with the Hamming cube. More generally, the paper organizes orbits of vertices and applies tools from Bass–Serre theory to build the countable graphs that reflect each group’s coarse geometry.