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Conformal dimension bounds for certain Coxeter group Bowditch boundaries

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Pith's one-line read For every large-type complete-graph Coxeter group with $m \ge 11$ vertices, the Bowditch boundary has conformal dimension at least $1 + \frac{\log(\lfloor (m-5)/3 \rfloor)}{\log(2M-1)}$.

desk verdict Strong and genuinely new lower bounds plus a plausible CAT(-1) upper-bound construction, but the deferred quasi-isometry proof in Proposition 4.9 is load-bearing and needs to be supplied before the upper bound is proved. read the letter →

arxiv 2504.12404 v1 pith:ZG547OEB submitted 2025-04-16 math.GT math.GR

Elizabeth Field , Radhika Gupta , Robert Alonzo Lyman , Emily Stark This is my paper · ORCID
classification math.GTmath.GR MSC 20F5520F6557M07
keywords conformaldimensionBowditchboundaryCoxetergroupsrelativelyhyperbolicroundtreesCAT(-1)spacesquasi-isometryclassesHausdorff
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The conformal dimension of a metric space is the smallest Hausdorff dimension one can obtain by deforming the space with controlled quasisymmetric distortions; for group boundaries it is a quasi-isometry invariant. This paper studies the Bowditch boundary of a large-type Coxeter group whose defining graph is a complete graph with all edge labels at least three, giving the first nontrivial bounds on its conformal dimension in a non-hyperbolic relatively hyperbolic setting. The lower bound, for $m \ge 11$ vertices with maximal edge label $M$, is $\operatorname{Confdim}(\partial(W_\Gamma,\mathcal{P})) \ge 1 + \frac{\log(\lfloor (m-5)/3 \rfloor)}{\log(2M-1)}$, obtained by embedding Gromov round trees into the Davis--Moussong complex while avoiding flats. The upper bound, roughly $13 + 12\log m + 19\log M$ for $M \ge 4$ and $23 + 12\log m$ for $M = 3$, comes from constructing a CAT($-1$) model space and estimating the Hausdorff dimension of its visual boundary. The paper concludes from these bounds that each family with a uniform bound on edge labels contains infinitely many quasi-isometry classes, and that related hyperbolic groups with Pontryagin sphere boundary have infinitely many quasi-isometry classes while their conformal dimensions accumulate densely in $(1,\infty)$.

What carries the argument

Two constructions carry the argument. A combinatorial round tree is a polygonal 2-complex built from $V$ copies of a half-plane-like piece glued in a rooted-tree pattern; its boundary is a Cantor set times an interval, and its conformal dimension is at least $1 + \frac{\log V}{\log H}$ when it embeds quasi-isometrically in a hyperbolic polygonal complex. The paper grows such a tree inside the Davis--Moussong complex, using a wall-crossing criterion to ensure that each stage's 1-skeleton is convex and periodically disallowing triples of edge labels so the tree has uniformly bounded intersection with flats. The upper-bound machinery is a CAT($-1$) model space $Y_\Gamma$: truncated blocks from the ideal regular tetrahedron in $\mathbb{H}^3$ are glued along kite faces and capped consistently with Euclidean or hyperbolic triangle subgroups; Gromov's link condition, verified through spherical joins and a metric-flag argument, makes the space CAT($-1$), and the visual metric with parameter $e$ on its boundary is the metric in which the Hausdorff dimension computation is performed.

What would settle it

A direct calculation that would settle the upper-bound chain: take the complete graph on four vertices with mixed edge labels, for instance $(3,3,4)$, construct the model space $Y_\Gamma$, and compare distances between disjoint prisms in $Y_\Gamma$ with the corresponding distances in the cusped Cayley graph along elements that alternate between the two triangle types; if the ratio of the two distances is unbounded, the asserted quasi-isometry fails and the upper-bound transfer does not hold.

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Extended reading notes

Core claim

The central claim is that for every complete defining graph with $m \ge 11$ and edge labels $m_{ij} \ge 3$, the Bowditch boundary of the relatively hyperbolic pair $(W_\Gamma,\mathcal{P})$ has conformal dimension at least $1 + \frac{\log(\lfloor (m-5)/3 \rfloor)}{\log(2M-1)}$, where $M = \max m_{ij}$, and at most $13 + 12\log m + 19\log M$ when $M \ge 4$ (with the special value $23 + 12\log m$ when $M = 3$). The lower bound is proved by constructing a combinatorial round tree with vertical branching $V = \lfloor (m-5)/3 \rfloor$ and horizontal branching $H = 2M - 1$ inside the Davis--Moussong complex; convexity is maintained one skeleton at a time, and a periodic forbidding of label triples keeps the tree from fellow-traveling with flats, so the tree boundary embeds in the Bowditch boundary. The upper bound is proved by building a CAT($-1$) space $Y_\Gamma$ from truncated blocks of the ideal regular tetrahedron in hyperbolic 3-space, checking Gromov's link condition, and applying the theorem that Hausdorff dimension of the conical limit set equals the critical exponent of the Poincar\'e series; orbit counts are then bounded through an itinerary-type decomposition of geodesics. The paper derives from these bounds infinitely many quasi-isometry classes in each family with bounded edge labels, infinitely many quasi-isometry classes among hyperbolic groups with Pontryagin sphere boundary, and, combined with an existing hyperbolic upper bound, a dense set of attainable conformal dimension values in $(1,\infty)$.

Load-bearing premise

The upper bound rests on the claim that the CAT($-1$) model space and the cusped Davis complex are equivalent at large scales; the proof is cited as a direct extension of an earlier construction with the details left to the reader, so if that claim fails the Hausdorff-dimension estimate cannot be transferred to the Bowditch boundary.

Editorial extensions

If this is right

  • All large-type Coxeter groups on complete graphs with a uniform ceiling on edge labels fall into infinitely many quasi-isometry classes, so Bowditch boundary topology alone cannot classify these groups.
  • Among hyperbolic Coxeter groups whose boundary is the Pontryagin sphere, there are infinitely many quasi-isometry classes.
  • For hyperbolic groups in this family, the conformal dimension of the boundary takes a dense set of values in $(1,\infty)$, found by matching the new lower bound with the existing hyperbolic upper bound.
  • In the all-labels-three family, any embedding into a truncated real hyperbolic space with polynomial distortion must have ambient dimension tending to infinity as the number of generators grows.
  • The lower and upper bounds are not sharp enough to complete the classification, which the authors conjecture is simply isomorphism.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The round-tree construction is not tied to Coxeter specifics: any CAT(0) group with isolated flats in which one can grow a convex tree with controlled intersection with flats should admit lower bounds of the form $1 + \frac{\log V}{\log H}$ on its relative boundary.
  • The cutoff $m \ge 11$ is likely removable; the authors explicitly expect round trees in the omitted small cases, and adapting the initial block should extend the lower bound to fewer vertices.
  • The upper-bound model is highly singular, and the authors state an expectation that its Hausdorff dimension grows with $M$ for fixed vertex count even while the true conformal dimension should decrease; if that expectation is right, the sharp boundary metric must come from a different construction than the CAT($-1$) model built here.
  • The most direct next step for the upper bound is to write out the deferred quasi-isometry proof for mixed edge labels; until then, the lower-bound theorem stands on a complete proof while the upper-bound theorem carries a stated gap.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies Coxeter groups WΓ whose defining graph is a complete graph on m vertices with all edge labels mij ≥ 3. For each such group, with P the collection of stabilizers of flats in the Davis–Moussong complex, the paper proves a lower bound (Theorem A / Theorem 3.8): Confdim(∂(WΓ,P)) ≥ 1 + log(⌊(m−5)/3⌋)/log(2M−1) for m ≥ 11, by constructing quasi-isometrically embedded combinatorial round trees and applying Mackay's conformal dimension estimate. It then constructs a CAT(−1) model space YΓ quasi-isometric to the cusped Davis complex (Theorem C / Theorem 4.1) and, via Paulin's critical exponent theorem and a counting argument, obtains upper bounds (Theorem B / Corollary 5.12): Confdim ≤ 23 + 12 log m if M = 3, and ≤ 13 + 12 log m + 19 log M if M ≥ 4. The paper derives corollaries on infinitely many quasi-isometry classes within families with bounded labels (Corollary 1.1), on hyperbolic groups with Pontryagin sphere boundary (Theorem 2.22), and on density of conformal dimension values (Corollary 3.9).

Significance. The results, if fully justified, would be a substantial contribution: they give the first nontrivial bounds on conformal dimension of the Bowditch boundary for non-hyperbolic relatively hyperbolic pairs, and they yield new quasi-isometry classification consequences for a well-studied family of Coxeter groups. The lower-bound construction is genuinely parameter-free: no constant is fitted to match a target dimension, and the upper-bound computation is explained in enough detail to be checked. The CAT(−1) model construction is intricate and, where worked out, the link-condition verification is explicit. The main caveat is that the paper's upper bounds rest on an equivariant quasi-isometry whose proof is explicitly deferred, and the lower bound relies on an angled-complex verification that is only sketched; these points are load-bearing but appear fixable.

major comments (4)
  1. [§4.1 (Proposition 4.9)] Proposition 4.9 is the essential bridge that transfers the Hausdorff-dimension computation on the CAT(−1) boundary ∂YΓ to the Bowditch boundary ∂(WΓ,P), and its proof is omitted. The proof refers to Cannon–Cooper [CC92, Section 4.2] and then says 'Their argument directly extends... We leave the details to the reader.' The cited case is the 4-generator group with all edge labels equal to 3, whereas the present construction has arbitrary complete graphs, varying edge labels, non-manifold gluings of truncated blocks, compact caps, and horoballs for Euclidean triangle subgroups. The asserted equivariant quasi-isometry must coarsely identify the horoball, thick, and hyperbolic-triangle pieces of YΓ with the corresponding pieces of the cusped Davis complex; this is not a routine formality. Since Theorem 5.11 and Corollary 5.12, and hence Theorem B, depend on this quasi-isometry, a complete proof is required before the upper bounds can be accepted.
  2. [§3.3 (Lemma 3.6)] Lemma 3.6 is a load-bearing step for Theorem A: it asserts that the round tree A is δ-hyperbolic, which is then used in Lemma 3.7 to embed ∂A into ∂(WΓ,P). The supplied proof via Blufstein–Minian is not complete. It asserts without justification that the subdivided complex A′ is 'simply connected and a flag (hence 3-flag)'; flagness is not immediate for a complex obtained by subdividing polygons in strip patterns, and no argument is given that every clique in the 1-skeleton is filled by a simplex. It also asserts that scaling the angles at the internal vertex u by 3/4 preserves 2π-largeness of the links; the discussion only addresses cycles in the link of u and does not check all vertices whose links contain scaled corners. These points need to be proved rather than left to the reader.
  3. [§3.2–3.3 (IH4 and Lemma 3.7)] Lemma 3.7 asserts that A quasi-isometrically embeds in the cusped Cayley graph X(WΓ,P) because 'the diameter of the intersection of any horoball with A is uniformly bounded by Induction Hypothesis (IH4).' This is exactly the point that prevents flats from creating shortcuts, and it is not proved. The paragraph after Construction 3.3 states that periodically disallowing each triple of labels ensures 'uniformly bounded intersection with every flat,' but no quantitative argument is given, nor is it explained why a flat cannot reappear repeatedly along the round tree. I request an explicit proof of the uniform bound, since without it ∂A need not embed in the Bowditch boundary.
  4. [§4.2.2 (Lemma 4.17)] Lemma 4.17 is central to the CAT(−1) verification of YΓ, but two steps in its proof are only asserted. In the treatment of the northern complex N, it is claimed that because each edge s_j^i is convex, an isometrically embedded circle crosses each such edge exactly once; the conclusion 'hence crosses each such edge exactly once' does not follow from convexity alone without checking possible multiple crossings or circles contained in the union of edges. In the treatment of S, the π-convexity of the subsets s_1^1 ∪ s_2^r is argued by saying 'so γ is contained in ρ as desired,' which is an unproved geometric assertion. Since Lemma 4.17 feeds into Proposition 4.19 and Theorem 4.23, these arguments should be written out in full.
minor comments (5)
  1. [§3.3 (Theorem 3.8)] In the proof of Theorem 3.8, the sentence 'applying Theorem 3.2 in the setting when the hyperbolic polygonal 2-complex is the round tree A itself' is confusing; one should state that Theorem 3.2 is applied with X = A and the identity embedding.
  2. [§5.2 (Theorem 5.9)] In Theorem 5.9, the distance d(P,P′) in conclusions (2) and (3) should be dYΓ(P,P′) to match Definition 5.4; the intended meaning is clear from context.
  3. [§5.5.3 (Lemma 5.20)] The term XΓ-hexagon is used before it is defined; a short definition, parallel to the definition of an XΓ-polygon in Notation 5.1, would improve clarity.
  4. [§5.4 (Corollary 5.12)] In Corollary 5.12, the estimates 'one sees' for A ≤ 3m^4M^2, B ≤ 20m^3M, and C1 ≤ 4m^3M^3 are used to produce the final constants; since these inequalities are not immediate, a short verification or appendix entry would improve the exposition.
  5. [§4.1.2 (Lemma 4.20)] In Lemma 4.20 the bound is stated as h ≤ sqrt(y0^2 + 1) for all cap types; for the hexagon cap the sharper bound sqrt(y0^2 + 1/3) holds, and the stated weaker bound is sufficient, but this could be noted to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: lower and upper bounds are derived from external theorems and explicit constructions; Proposition 4.9's deferred proof is a correctness gap, not a circular step.

full rationale

No significant circularity found. The lower-bound chain (Theorem A) uses Mackay's round-tree theorem as an external input and constructs a combinatorial round tree inside the Davis complex with convexity and flat-avoidance proved in Lemmas 3.6 and 3.7. The branching data V = floor((m-5)/3) and H = 2M-1 are read off the construction, not fitted to the target conformal dimension. The upper-bound chain (Theorem B) builds an explicit CAT(-1) model YΓ, applies Paulin's critical exponent theorem, and obtains explicit counting estimates for orbit points (Theorem 5.9, Proposition 5.10). The constant y0 = 1.5 is chosen only to satisfy the link-condition inequalities in Proposition 4.19 and the Appendix, not to match a desired conformal dimension value. No load-bearing self-citations appear; the main external inputs are independent theorems of Mackay, Bourdon-Kleiner, Paulin, and Cannon-Cooper. The only flagged concern is Proposition 4.9, where the equivariant quasi-isometry between YΓ and the cusped Cayley graph is asserted and its proof deferred: 'Their argument directly extends... We leave the details to the reader.' This is an omitted proof and a genuine correctness risk, but it is not circularity: the quasi-isometry is a stated premise used to transfer a dimension bound, and there is no quoted equation or construction showing that the target conformal dimension is fed back as an input. Therefore the derivation chain does not reduce to its own assumptions, and the circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The free-parameter count is minimal: the only hand-chosen constant is the horosphere height y0=1.5, which is a construction parameter that satisfies the explicit link-condition inequalities rather than a fitted value. The axioms are standard results in the field, cited and applied where needed. No new entities are postulated; the CAT(-1) model is assembled from standard hyperbolic blocks.

free parameters (1)
  • y0 (horosphere height) = 1.5
    Height of the horosphere used in constructing the CAT(-1) model space; chosen so the cap vertex link condition holds (Proposition 4.19). The value 1.5 is an explicit solution to the inequalities, and the upper bound constants depend on it.
assumptions (5)
  • standard math The Davis-Moussong complex of a Coxeter group is CAT(0) (Moussong's theorem).
    Used throughout Section 3 to construct the round tree in a CAT(0) complex and to apply convexity of walls.
  • domain assumption For relatively hyperbolic group pairs whose peripheral subgroups are not nontrivially relatively hyperbolic, the quasisymmetry type of the Bowditch boundary is a quasi-isometry invariant (BDM09, Gro13, MS24, HH).
    This is the basis for Corollary 2.11 and for using conformal dimension as a quasi-isometry invariant; the peripheral subgroups here are flat stabilizers.
  • standard math Mackay's round tree theorem (Theorem 3.2) gives a conformal dimension lower bound from a quasi-isometrically embedded combinatorial round tree in a hyperbolic complex.
    Applied in Theorem 3.8 with the round tree A itself as the hyperbolic polygonal complex.
  • standard math Paulin's theorem (Theorem 2.16) equates the critical exponent of a CAT(-1) group action with the Hausdorff dimension of the conical limit set with visual metric parameter e.
    Used in Theorem 5.11 to convert the orbit count into the Hausdorff dimension upper bound.
  • standard math Bourdon-Kleiner's upper bound for hyperbolic Coxeter groups (BK15, Corollary 8.1) is used for the density application.
    Combined with the new lower bound in Corollary 3.9 to show conformal dimension is dense.

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Pith. "Pith review of Conformal dimension bounds for certain Coxeter group Bowditch boundaries." pith.science (2026). https://pith.science/paper/ZG547OEB

@misc{pith2026250412404,
  author       = {Pith},
  title        = {Pith review of: Conformal dimension bounds for certain Coxeter group Bowditch boundaries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZG547OEB}},
  note         = {Machine review of arXiv:2504.12404}
}
abstract

We give upper and lower bounds on the conformal dimension of the Bowditch boundary of a Coxeter group with defining graph a complete graph and edge labels at least three. The lower bounds are obtained by quasi-isometrically embedding Gromov's round trees in the Davis complex. The upper bounds are given by exhibiting a geometrically finite action on a CAT(-1) space and bounding the Hausdorff dimension of the visual boundary of this space. Our results imply that there are infinitely many quasi-isometry classes within each infinite family of such Coxeter groups with edge labels bounded from above. As an application, we prove there are infinitely many quasi-isometry classes among the family of hyperbolic groups with Pontryagin sphere boundary. Combining our results with work of Bourdon--Kleiner proves the conformal dimension of the boundaries of hyperbolic groups in this family achieves a dense set in $(1,\infty)$.

Figures

Figures reproduced from arXiv: 2504.12404 by the authors.

Figure 3.1
Figure 3.1. The 2-complex Aan in the case that mij = 3 and the 2-cells in the Davis–Moussong complex are hexagons. The initial complex A0 is the union of the three hexagons at the top of the figure, and its outer edge path E0 is drawn in red. The outer edge path of Aan is Ean , which is drawn in red, and its adjacent path Ean−1 is drawn in orange. The internal vertices v1, . . . , vk along Ean are indicated with black circles. … view at source ↗
Figure 3.2
Figure 3.2. Gluing in polygons during the inductive step. The red line depicts a segment of the outer edge path Ean . in each strip subcomplex S j ak with j satisfying 1 ≤ j ≤ V are not labeled with a pair in the triple tℓ, where ℓ = k mod R. (By periodically disallowing every triple of generators to repeat, we ensure the round tree has uniformly bounded intersection with every flat.) Inductive Step. To construct the complex An… view at source ↗
Figure 3.3
Figure 3.3. Configurations of walls locally. and Pq in An+1, respectively. If Pp and Pq intersect non-trivially, then one can verify that a geodesic in X (1) Γ is contained in the 1-skeleton of these polygons, and is hence contained in A (1) n+1. Henceforth, we suppose Pp and Pq are disjoint. Let p¯ and q¯ denote the vertices of A (1) n nearest to p and q respectively. Since Pp and Pq intersect A (1) n , we have p¯ ∈ Pp and q¯ … view at source ↗
Figures from the paper (10 more)
Figure 4.1
Figure 4.1. Figure 4.1: On the left a truncated block is the indicated subspace of the ideal tetrahedron T with boundary a triangle, three pentagonal faces, and three shaded kite faces. These kites correspond to the three shaded kites in the portion of the Davis complex illustrated on the r…
Figure 4.2
Figure 4.2. Figure 4.2: On the left is a regular tiling of the 2-sphere by triangles with angles 2π 3 . On the right is a labeled side of the ideal tetrahedron T viewed in the upper half space model of the hyperbolic plane. Hausdorff dimension of its boundary. We note that the value y0 = 1.…
Figure 4
Figure 4. Figure 4: highlights three Euclidean kites in the Davis complex. Note that the isometry type of each [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 4.3
Figure 4.3. Figure 4.3: On the left is the Davis complex for ∆(3, 3, 4) and its dual tiling by triangles. On the right in blue is the alternative cellulation by equilateral Euclidean triangles with the key property that the link of every vertex has cone angle strictly greater than 2π. In th…
Figure 4.4
Figure 4.4. Figure 4.4: The three isometry types of the caps used to build the space YΓ. in the metric cone C−1(Lk(v, K)) (see [BH99, Definition I.5.6]). Now Berestovskii’s theorem [BH99, Theorem 3.14] implies that C−1(Lk(v, K)) is CAT(−1) if and only if Lk(v, K) is CAT(1). Next suppose x ∈…
Figure 4.5
Figure 4.5. Figure 4.5: The spherical complex Lθ,2r built from 2r = 8 spherical isosceles triangles with two sides of length θ for π 2 < θ < π and with the angle between these sides equal to π 3 . We show the complex is CAT(1) provided θ is sufficiently close to π 2 so that the boundary loo…
Figure 4.6
Figure 4.6. Figure 4.6: The blue angle between hyperbolic geodesics indicated in the figure is relevant to building a CAT(−1) model geometry. Its relationship between the values h and D is computed in Lemma 4.18. isometrically embeds in S 2 and s 1 1 ∪ s 2 3 is contained in a great circle o…
Figure 4.7
Figure 4.7. Figure 4.7: Triangle tilings T1(F) of the boundary components of YˆΓ are shown in black. In red is the alternative tiling T2(F), by triangles on the left and hexagons on the right, so that each vertex in this tiling has link of cone angle strictly greater than 2π. The left pictu…
Figure 4.8
Figure 4.8. Figure 4.8: The red triangles are equilateral triangles on a horosphere in H 3 . The blue lines depict hyperbolic geodesics between their endpoints, and the blue shaded triangles lie on a halfspace containing the three red points. 5 Upper bounds on the conformal dimension In thi…
Figure 5.1
Figure 5.1. Figure 5.1: Computing the distance between a XΓ-polygon and a wall. We now show dYΓ (v, T) = dYΓ (v, W). Let πW : YΓ → W be the nearest point projection map to the convex wall W. We claim that πW (v) = u. Suppose not. Then there exists a geodesic triangle with vertex set {v, u, …

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