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arxiv: 2607.01196 · v1 · pith:Y3NPY6H5new · submitted 2026-07-01 · 🧮 math.GR · math.GN· math.GT

Coarse geometry of homeomorphism groups: Classifying countable Stone spaces

George Domat , Hannah Hoganson , Robert Alonzo Lyman This is my paper

Pith reviewed 2026-07-02 04:01 UTC · model grok-4.3

classification 🧮 math.GR math.GNmath.GT
keywords groupscoarselyboundedclassescoarsehammingcountableequivalence
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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.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors work with homeomorphism groups of countable Stone spaces, which are topological groups that are not locally compact. In earlier work they divided these groups into three types based on whether they are coarsely bounded, unbounded but generated by a coarsely bounded set, or unbounded without such a generating set. The new result is that these three types are in fact the complete list of coarse equivalence classes: any two groups of the same type are coarsely equivalent. Groups of the middle type turn out to be quasi-isometric to the Hamming cube, the space of infinite binary sequences with finitely many 1s under Hamming distance. As a side result they prove that Hamming graphs on finite alphabets are all bi-Lipschitz equivalent when infinite. The work is part of an effort to develop coarse geometry for groups that do not fit the usual locally compact setting.

Core claim

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... infinite Hamming graphs over finite alphabets are all bi-Lipschitz equivalent.

Load-bearing premise

The three-class partition obtained in the authors' previous paper is taken as given and the new arguments show that this partition coincides exactly with the coarse equivalence relation; if the prior partition missed a class or over-split, the current classification claim would fail.

Figures

Figures reproduced from arXiv: 2607.01196 by George Domat, Hannah Hoganson, Robert Alonzo Lyman.

Figure 1
Figure 1. Figure 1: A small piece of the graph Γ. The partitions [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A piece of the graph Γ realized as the poset graph of finite subsets of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A piece of the graph Γ realized as the countably infinite Hamming cube. This shows the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A schematic of the Bass-Serre T together with the quotient ray of groups G\T as in Proposition 5.1. In the tree T, all of the bottommost vertices are indexed via cosets of G1, then the next are indexed via cosets of G2, etc. on the vertex set of T˜ where h ∈ H. We now have a bijection between the vertices of Γ and the ˜ set G/H × N and we will make use of this labeling of the vertices. We refer to the edge… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Circularity Check

0 steps flagged

Minor self-citation for partition; new equivalence proofs independent

full rationale

The paper cites its own prior work solely to obtain the three-class partition of Homeo(X) groups by coarse-boundedness properties. The present arguments then establish that groups inside each class are coarsely equivalent (and quasi-isometric to the Hamming cube for the middle class) via new constructions and comparisons that do not reduce by definition or by construction to the cited partition. No equation or step inside the current derivation is shown to be equivalent to its inputs; the self-citation supplies the classes but is not load-bearing for the equivalence or quasi-isometry claims themselves. This is a normal, non-circular use of prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard facts from coarse geometry, topological group theory, and the authors' own prior classification; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard definitions and basic properties of coarse equivalence, quasi-isometry, and coarsely bounded sets in topological groups.
    Invoked implicitly when stating that the three classes coincide with coarse equivalence classes.
  • domain assumption The three-class partition of homeomorphism groups from the authors' previous paper.
    The current classification claim is built directly on this partition being exhaustive and correct.

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