Coarse geometry of homeomorphism groups: Classifying countable Stone spaces

Abstract: Towards developing the tools of geometric group theory for non-locally compact topological groups, we give one of the first complete classifications of a family of such groups up to coarse equivalence, and when possible, up to quasi-isometry. In a previous paper, we placed the homeomorphism groups of countable Stone spaces into three classes: coarsely bounded, unbounded yet generated by a coarsely bounded set, and unbounded but not generated by any coarsely bounded set. Now we show that these are the coarse equivalence classes: Any two groups within one of these classes are in fact coarsely equivalent. Furthermore, we show that groups in the second class are quasi-isometric to the Hamming cube, the space comprising infinite binary sequences with finitely many nonzero entries equipped with the Hamming distance. As part of the proof, we show that infinite Hamming graphs over finite alphabets are all bi-Lipschitz equivalent.

• The paper establishes that homeomorphism groups of countable Stone spaces fall into three distinct coarse equivalence classes based on Cantor–Bendixson rank and maximal point count.
• It uses explicit geometric models like Cayley–Abels–Rosendal graphs and trees of Hamming cubes, demonstrating a 16-bi-Lipschitz equivalence in the CB-generated successor case.
• The classification offers insights for extending coarse geometric analysis to non-locally compact groups and mapping class groups of infinite-type surfaces.

Coarse Classification of Homeomorphism Groups of Countable Stone Spaces

Introduction and Context

This paper provides a comprehensive coarse geometric classification of homeomorphism groups of countable Stone spaces, denoted $X_{\alpha, n}$, determined by two invariants: Cantor–Bendixson rank $\alpha$ (a countable ordinal) and the number $n \ge 1$ of maximal-rank points. The principal object of study is $\Homeo(X_{\alpha,n})$ endowed with the compact-open topology. The work achieves a rigorous determination of coarse equivalence and quasi-isometry classes within this family, establishing strong structural results in the context of geometric group theory for non-locally compact topological groups.

Main Classification Theorem

The fundamental result, Theorem A, establishes that the structure of $\Homeo(X_{\alpha,n})$ up to coarse equivalence falls into exactly three distinct classes, characterized completely by the parameters $\alpha$ and $n$:

1. Coarsely bounded: For $n=1$ and any $\alpha$, $\Homeo(X_{\alpha,1})$ is coarsely bounded and hence quasi-isometric to a point.
2. Unbounded but CB-generated (successor case): For $\alpha$0 and $\alpha$1 a successor ordinal, $\alpha$2 is quasi-isometric to the countably infinite Hamming cube—more precisely, to the graph of finite-support binary sequences with the Hamming metric.
3. Unbounded, not CB-generated (limit case): For $\alpha$3 and $\alpha$4 a limit ordinal, $\alpha$5 is coarsely equivalent to a level set in a “tree of Hamming cubes,” modeled on a 1-ended tree of infinite Hamming cubes with a prescribed branching structure.

The paper proves that these classes are sharp: any two groups in a class are coarsely equivalent, and the distinctions are robust under the parameters. These are among the first such complete coarse and quasi-isometric classifications for infinite families of non-locally compact Polish groups.

Methods and Geometric Models

Cayley–Abels–Rosendal Graphs

For CB-generated groups, explicit Cayley–Abels–Rosendal (CAR) graphs are constructed as geometric models capturing the quasi-isometry type. In the successor ordinal case, these CAR graphs are shown to be isomorphic to the (countably) infinite Hamming graph $\alpha$6, with vertices corresponding to partitions of $\alpha$7 and adjacency realized via finite “shifts.” A bijective correspondence is made between the group’s geometric model and $\alpha$8 with the Hamming distance.

Bi-Lipschitz Equivalence of Hamming Cubes

A significant technical accomplishment is the establishment that all $\alpha$9 (for $n \ge 1$0) are $n \ge 1$1-bi-Lipschitz equivalent, using an explicit combinatorial “back-and-forth” construction and a swindle employing the infinite product structure. This enables the identification of all quasi-isometry classes in the CB-generated (successor) regime with that of $n \ge 1$2.

Trees of Hamming Cubes and Non-CB-Generated Regime

For the limit ordinal case—when the group is locally CB but not CB-generated—the paper develops new graphical objects: trees of Cayley–Abels–Rosendal graphs, and, by quotienting, trees of Hamming cubes ($n \ge 1$3). The construction leverages an exhaustive chain of open subgroups and Bass–Serre theory to organize the group action on a 1-ended tree, then blows up vertices to fibers isomorphic to Hamming cubes. It is shown that $n \ge 1$4 is coarsely equivalent to the set of minimal-height vertices in $n \ge 1$5.

These graphical models’ fibers within a level (and, by extension, the models as a whole for different $n \ge 1$6) are shown to be bi-Lipschitz equivalent, thus yielding precise coarse equivalences.

Stone Space, Cantor–Bendixson Invariants, and Generating Sets

Countable Stone spaces are uniquely classified by $n \ge 1$7: the Cantor–Bendixson derivative yields a filtration with each step removing isolated points, and maximal points correspond to those surviving at the last non-empty derivative. The paper systematically analyzes the action of $n \ge 1$8 on partitions and points of various ranks, leading to an explicit description of coarse boundedness and the kernel projections that underpin the main coarsening reduction: all CB-generated homeomorphism groups reduce quasi-isometrically to the “base” case $n \ge 1$9.

Connections to Discrete Topology, Mapping Class Groups, and Graphs

Results translate, via a dual perspective, to groups with the discrete topology, where the notion of strong boundedness aligns with coarse boundedness; the SB-generated groups are again quasi-isometric to $\Homeo(X_{\alpha,n})$0. This refines and subsumes results regarding infinite symmetric groups and other strongly bounded group classes.

The findings have implications for the study of mapping class groups (MCG) of infinite-type surfaces, whose action on the space of ends (a countable Stone space) fits into a short exact sequence with $\Homeo(X_{\alpha,n})$1 and $\Homeo(X_{\alpha,n})$2. The classification here demonstrates that quasi-isometry type distinctions in MCGs (beyond those forced by end structure) must arise from deeper topological features of the surfaces. This contrasts with group mapping class groups of graphs, which, for infinite trees, exhibit an exact parallel with the classification for Stone spaces.

The investigation identifies precise conditions under which the forgetful maps or induced maps on end/cantor–bendixson invariants do not yield coarse equivalence in the MCG context, emphasizing that purely “combinatorial” end-space data is insufficient for quasi-isometric rigidity for these more geometric groups.

Numerical and Structural Highlights

• All unbounded, CB-generated homeomorphism groups of countable Stone spaces $\Homeo(X_{\alpha,n})$3, $\Homeo(X_{\alpha,n})$4, $\Homeo(X_{\alpha,n})$5 successor, are quasi-isometric to each other and to the infinite Hamming cube ($\Homeo(X_{\alpha,n})$6), with a universal bi-Lipschitz constant $\Homeo(X_{\alpha,n})$7.
• In the non-CB-generated (limit ordinal) case, the group is always coarsely equivalent to a level set in a canonical tree of Hamming cubes, again with the geometric model independent of $\Homeo(X_{\alpha,n})$8.
• Explicit construction of the geometric models (CAR graphs or tree-of-Hamming-cubes) yields effective representatives for these coarse equivalence classes.

Implications and Outlook

This classification supplies a new foundation for the systematic use of coarse geometric tools in non-locally compact Polish groups, providing clean families where all necessary invariants and representatives are explicit. The construction and results strongly suggest that similar trichotomies will appear in other families of non-locally compact transformation groups, and the techniques (particularly tree-of-space constructions) are generalizable.

For geometric group theory, the result delineates a clear boundary: in the absence of local compactness and finite generation, coarse geometry can still be rich and tractable, but also highly structured, with many a priori distinct objects collapsing to a handful of geometric types.

Future developments may include:

• Systematic study of trees (or more general graphs) of geometric models for other non-locally compact groups.
• Refined quasi-isometric invariants for mapping class groups of infinite-type surfaces—beyond those accessible via end spaces alone.
• Generalization of tree-of-CAR-graph methodology to contexts such as automorphism groups of other profinite structures, or in the setting of descriptive set theory-invariant coarse geometry.

Conclusion

This paper realizes a thorough coarse geometric classification of homeomorphism groups of countable Stone spaces, demonstrating that they distribute into exactly three coarse types, distinguished by Cantor–Bendixson rank and the number of maximal points. The work combines group-theoretic, combinatorial, and topological arguments with explicit geometric and combinatorial models, resolving a foundational question in the coarse geometry of non-locally compact transformation groups and opening avenues for further exploration of nondiscrete, infinite symmetry.