{"id":"f3ada9a1-a4d6-4206-8ce3-1eded0df5900","arxiv_id":"2504.12404","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Large-type Coxeter groups on complete graphs have Bowditch boundary conformal dimension bounded by roughly 1+log(m)/log(2M) from below and 13+12log m+19log M from above, separating them into infinitely many quasi-isometry classes.","lead":"This paper proves upper and lower bounds on the conformal dimension of Bowditch boundaries for large-type Coxeter groups on complete graphs, and shows these groups split into infinitely many quasi-isometry classes. The bounds are the first of their kind for non-hyperbolic relatively hyperbolic group pairs, offering new tools to distinguish such groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Upper bound rests on Proposition 4.9, whose equivariant quasi-isometry is left to the reader; unless it is proved, Theorem B transfers a dimension bound from ∂YΓ to the Bowditch boundary without justification.","rationale":"I read the lower-bound construction, the CAT(−1) link verification, and the Hausdorff-dimension counting. The reader's weakest-assumption identification is accurate: Proposition 4.9 is the single most load-bearing gap because every upper-bound conclusion flows through it. The paper's own text admits the proof is deferred, and the construction of YΓ is sufficiently intricate that 'directly extends' is not a formality. In particular, the cusped Davis complex has horoballs only over Euclidean triangle subgroups, while YΓ also has compact caps over hyperbolic triangle subgroups; an equivariant quasi-isometry must respect this coarse structure, and no argument is supplied. The lower-bound direction appears more secure: the round tree construction is explicit, and Proposition 3.5 plus Lemma 3.6 give a plausible route, although Lemma 3.6's assertion that the subdivided complex is flag is terse. Still, that gap is smaller and more likely repairable than the omitted quasi-isometry. I therefore agree with the CONDITIONAL verdict: the upper bound should not be accepted until the deferred proof of Proposition 4.9 is supplied, and the flagness point in Lemma 3.6 deserves a clarifying sentence. No change to the reader's verdict is needed.","tokens_in":43613,"tokens_out":15058,"duration_ms":172088,"concrete_test":"Write out the Cannon–Cooper equivariant quasi-isometry explicitly for the minimal case where the extension is nontrivial: Γ = K4 with edge labels (3,3,3,4), so one triangle subgroup is Euclidean and one is hyperbolic. Construct the map YΓ → X(WΓ,P), verify equivariance and coarse bi-Lipschitz surjectivity, and check that the compact cap over the ∆(3,3,4) boundary plane lands within a uniformly bounded neighborhood of the corresponding Davis subcomplex. If this special case cannot be proved, Proposition 4.9—and hence Theorem B—is unsupported; if it can be proved, the same argument should be made explicit for general Γ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem B and Theorem C depend on Proposition 4.9, which asserts that the CAT(−1) space YΓ is equivariantly quasi-isometric to the cusped Davis complex X(WΓ,P). The proof cites Cannon–Cooper [CC92, Section 4.2] and then says: 'Their argument directly extends... We leave the details to the reader.' This is an omitted proof of a load-bearing premise. The extension is not a routine formality: [CC92] treats the all-edge-label-3 K4 case, whereas the present YΓ has varying edge labels, non-manifold gluings around kites, compact hyperbolic caps attached to hyperbolic triangle subgroups, and horoball cusps for Euclidean triangle subgroups. The cusped Davis complex attaches combinatorial horoballs only to flat stabilizers, so a quasi-isometry must correctly coarsely identify which pieces of YΓ correspond to horoballs, to thick parts, and to hyperbolic triangle subcomplexes. If this quasi-isometry fails, Theorem 2.5 and Corollary 2.11 cannot transfer the Hausdorff-dimension upper bound computed on ∂YΓ to the Bowditch boundary ∂(WΓ,P), and Theorem B would not follow. This concern is independent of the lower-bound construction and is explicitly flagged in the manuscript as a deferred proof. The remainder of the upper-bound computation is lengthy but appears internally consistent; however, it becomes relevant only if Proposition 4.9 is true.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":43849,"tokens_out":15323,"duration_ms":153150,"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":[{"comment":"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.","section":"§4.1 (Proposition 4.9)"},{"comment":"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.","section":"§3.3 (Lemma 3.6)"},{"comment":"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.","section":"§3.2–3.3 (IH4 and Lemma 3.7)"},{"comment":"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.","section":"§4.2.2 (Lemma 4.17)"}],"minor_comments":[{"comment":"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.","section":"§3.3 (Theorem 3.8)"},{"comment":"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.","section":"§5.2 (Theorem 5.9)"},{"comment":"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.","section":"§5.5.3 (Lemma 5.20)"},{"comment":"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.","section":"§5.4 (Corollary 5.12)"},{"comment":"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.","section":"§4.1.2 (Lemma 4.20)"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the deferred proof of Proposition 4.9; if the authors can supply it, together with the missing arguments in Lemma 3.6 and Lemma 3.7, I would support publication. I see no sign of circularity: the lower and upper bounds are derived from independent tools (round trees and critical-exponent counting), and no fitted parameter is introduced to match the target inequalities. The choice y0 = 1.5 is made after the fact to satisfy a concrete inequality and does not affect the final constants."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious paper and it deserves a real referee, but I would not accept it as is. What is genuinely new is the lower-bound side: a round-tree construction embedded in the Davis–Moussong complex for large-type Coxeter groups, adapted to the non-hyperbolic relatively hyperbolic setting by avoiding flats. That gives the first nontrivial conformal dimension lower bounds for Bowditch boundaries of non-hyperbolic relatively hyperbolic pairs. The CAT(-1) model space built from truncated ideal tetrahedra is also new, and the upper-bound computation is a substantial piece of work. The main theorems are plausible, the strategy is coherent, and the claimed novelty relative to the literature holds up: previous round-tree applications were in hyperbolic settings, and there were no prior bounds like these.\n\nCredit where due: the lower bounds use Mackay's round tree theorem with no fitted parameters; H depends only on M and m. The application to infinite quasi-isometry classes, the Pontryagin sphere result, and the Bourdon–Kleiner density argument all follow cleanly once the bounds are in hand. The citation pattern looks fair, and the manuscript is honest about places where it expects the construction to be non-sharp.\n\nThe soft spots are real and load-bearing. Proposition 4.9 asserts that the constructed CAT(-1) space YΓ is equivariantly quasi-isometric to the cusped Davis complex, then says the argument directly extends from Cannon–Cooper and leaves details to the reader. That is not a routine formality: the Cannon–Cooper setting is the all-label-3 K4 case, while here edge labels vary, non-manifold gluings occur around kites, caps are attached to hyperbolic triangle subgroups, and horoballs are attached to Euclidean ones. Without this quasi-isometry, the Hausdorff dimension computation on ∂YΓ cannot be transferred to the Bowditch boundary, so Theorem B does not follow. I think the statement is likely true and repairable, but it has to be written out. Lemma 3.6 has a smaller but similar gap: the strictly systolic angled complex argument asserts flagness without proof, and the angle-scaling step at valence-4 vertices needs a more careful check.\n\nWho is this for? People working on conformal dimension, Coxeter groups, and quasi-isometry classification will get real value from it. Send it out for review; the referee should insist on a full proof of Proposition 4.9 and a cleaned-up Lemma 3.6 before acceptance.","headline":"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.","tokens_in":44418,"tokens_out":1836,"would_cite":true,"duration_ms":23123,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F55","20F65","57M07"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every large-type complete-graph Coxeter group with $m \\ge 11$ vertices, the Bowditch boundary has conformal dimension at least $1 + \\frac{\\log(\\lfloor (m-5)/3 \\rfloor)}{\\log(2M-1)}$.","keywords":["conformal dimension","Bowditch boundary","Coxeter groups","relatively hyperbolic groups","round trees","CAT(-1) spaces","quasi-isometry classes","Hausdorff dimension"],"falsifier":"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.","tokens_in":43372,"feed_emoji":"📐","tokens_out":13946,"duration_ms":139650,"temperature":0.7,"pith_summary":"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 $(1,\\infty)$.","feed_headline":"Coxeter boundary dimension pinned between two log formulas","feed_subtitle":"Boundary-dimension bounds yield infinitely many quasi-isometry classes in every large-type Coxeter family.","key_machinery":"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.","core_discovery":"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 $(1,\\infty)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the round-tree estimate $1 + \\frac{\\log V}{\\log H}$ that converts branching data into the lower bound on conformal dimension.","marker":"[Mac16]"},{"why":"Introduces round trees; the paper embeds one into the Davis complex to produce a Cantor-set factor in the boundary.","marker":"[Gro93]"},{"why":"Supplies the Davis--Moussong complex, its CAT(0) metric, and the wall and carrier convexity used to keep the round tree convex.","marker":"[Dav08]"},{"why":"Gives the visual metric with parameter $e$ on the boundary of a CAT($-1$) space, the metric in which the upper-bound Hausdorff dimension is measured.","marker":"[Bou95b]"},{"why":"States that the critical exponent of a discrete isometry group equals the Hausdorff dimension of its conical limit set, transferring orbit counts to dimension bounds.","marker":"[Pau97]"},{"why":"Carries the deferred proof of the CAT($-1$) model's quasi-isometry to the cusped complex in Proposition 4.9, asserted to be a direct extension of this paper's argument.","marker":"[CC92]"},{"why":"Shows the augmented Cayley graph is equivariantly quasi-isometric to the cusped Cayley graph, completing the chain that lets $\\partial Y_\\Gamma$ stand in for the Bowditch boundary.","marker":"[GMS19]"},{"why":"Provides the hyperbolic-family upper bound that, combined with Theorem 3.8, yields density of conformal dimension values in $(1,\\infty)$.","marker":"[BK15]"},{"why":"Establishes that hyperbolic Coxeter groups with surface nerve have Pontryagin sphere boundary, the setting of the infinitely-many-quasi-isometry-classes application.","marker":"[\\u01520]"}],"fun_headline_variants":[],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"error":"Client error '402 Payment Required' for url 'https://api.deepseek.com/chat/completions'\nFor more information check: https://developer.mozilla.org/en-US/docs/Web/HTTP/Status/402"},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:32:30.180720+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}