{"id":"727e7770-576f-4c13-8923-e68fad171e05","arxiv_id":"2509.04323","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Torsion-free relatively hyperbolic groups with non-relatively-hyperbolic peripheral subgroups are finite index rigid, and peripheral-structure-preserving isomorphisms between finite index subgroups force equal indices.","lead":"This paper proves that, for many groups built from hyperbolic geometry, any two isomorphic finite-index subgroups must have the same index in the ambient group. The result covers fundamental groups of finite-volume negatively curved manifolds, limit groups, and free-by-cyclic groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.1 is applied in §7 with only a continuous boundary extension, not the required boundary homeomorphism; without bijectivity the cohomological pushforward fails, so the lower index bound is not established.","rationale":"The reader's weakest assumption exactly identifies the same load-bearing concern: the boundary map ∂Φ is only shown to be continuous, while Theorem 6.1 requires a homeomorphism. This is the single most critical point because all of Theorem 1.4's lower bound—and hence Theorem 1.1—rests on the uniform quasi-surjectivity stated in Theorem 6.1. If the boundary homeomorphism is not established, the contradiction argument in Theorem 6.1 fails and the complexity lower bound collapses. I agree with the reader that this is a fixable gap rather than a fatal flaw: one could likely choose Φ0 as a quasi-isometry mapping cusps to cusps, which would give a quasi-isometry on the cusped spaces and hence a boundary homeomorphism, or one could strengthen the cohomological argument to work with only a continuous map that has a nonzero induced map on the relevant Čech cohomology group. But as written, the proof is incomplete. I therefore see no reason to change the reader's CONDITIONAL verdict. I considered whether the arbitrary Φ0 might also violate Lemma 6.3's quasi-isometry hypothesis; that compounds the concern, but the cleanest statement remains the missing bijectivity. The paper otherwise benefits from strong external ingredients (Mineyev bicombing, Manning–Wang cohomology, Dahmani's classifying spaces), so the overall strategy is credible.","tokens_in":18961,"tokens_out":9692,"duration_ms":95766,"concrete_test":"Re-examine §7 Step 3 and verify whether the hypotheses of Lemma 6.3 and Theorem 6.1 are simultaneously satisfied. Concretely: check whether the construction can be modified so that Φ0 is an H-equivariant quasi-isometry preserving the peripheral structure (so Lemma 6.3 applies) and then prove that the induced boundary map is injective—for example, by showing that distinct parabolic points in ∂Ccyl(K) map to distinct parabolic points in ∂Ccyl(X) because Φ0 maps each peripheral subgroup quasi-isometrically onto its image. If such a proof cannot be supplied, test the theorem on a concrete example: let G be a non-elementary relatively hyperbolic group whose quotient relative classifying space has at least two vertex orbits, choose H=G, and take Φ0 that collapses two vertex orbits; compute the induced boundary map and check whether it is bijective. If it is not, Theorem 6.1 as stated cannot be","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central Theorem 1.4 depends on the uniform quasi-surjectivity of the map Φ constructed in §7. Theorem 6.1 explicitly requires that Φ extends continuously to a homeomorphism ∂Φ: ∂Ccyl(K,B) → ∂Ccyl(X,A). In §7 Step 3, the authors write 'By Lemma 6.3, Φ extends continuously to a map ∂Φ. It now follows from Theorem 6.1...' But Lemma 6.3 only proves continuity of the boundary extension; it says nothing about injectivity or surjectivity. The proof of Theorem 6.1 uses that (∂Φ)* is an isomorphism on Čech cohomology to pull back a nontrivial class and obtain a contradiction if Φ misses a ball. If ∂Φ is only continuous, (∂Φ)* need not be injective, so a nontrivial a could push forward to zero, and the contradiction fails. The paper gives no separate argument that the specific Φ constructed in §7—built from an arbitrary H-equivariant vertex map Φ0 that may not even be a quasi-isometry—induces a bijection on Bowditch boundaries. Without a homeomorphism, the conclusion X ⊆ N_{R0}(Φ(Ccyl(K))) does not follow from Theorem 6.1, and the lower bound α[G:H] ≤ C(H,PH) is unsupported. This is a genuine gap in the written proof, though likely fixable by choosing Φ0 more carefully or by weakening Theorem 6.1's hypothesis to a continuous boundary map with a nonzero induced cohomology map.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves finite index rigidity for torsion-free relatively hyperbolic groups whose peripheral subgroups are non-relatively-hyperbolic (NRH), and a more general rigidity statement for isomorphisms that preserve induced peripheral structures. The engine is a complexity invariant C(G,P), defined as the minimal number of cells in a simplicial relative classifying space. The authors prove a linear lower bound alpha[G:H] <= C(H,P_H) for finite-index H (Theorem 1.4) by combining weighted singular patterns from globally stable bicombings (Sections 3-4), cohomological uniform quasi-surjectivity for cylindrical cusped spaces (Sections 5-6), and a map-construction argument (Section 7). The upper bound C(H,P_H) <= beta[G:H] is a covering-space argument. A Reznikov-style liminf then converts these bounds into index rigidity.","tokens_in":19252,"tokens_out":9904,"duration_ms":100975,"significance":"If the proof is completed, the paper substantially extends the first author's finite-index rigidity theorem for hyperbolic groups to a broad relative setting, covering fundamental groups of finite-volume negatively curved manifolds, limit groups, and exponentially growing free-by-cyclic groups. The complexity invariant is natural and is defined independently of the target theorem; the linear bounds are derived rather than fitted. The proof strategy is credible and the paper is well organized. The main reservation is a missing verification of a boundary-homeomorphism hypothesis in Section 7; I regard the gap as likely repairable rather than fatal.","major_comments":[{"comment":"Theorem 6.1 is applied to conclude X subseteq N_{R0}(Phi(Ccyl(K))). Its hypothesis requires Phi to extend continuously to a homeomorphism dPhi: dCcyl(K,B) -> dCcyl(X,A). In Step 3 the authors write: 'By Lemma 6.3, Phi extends continuously to a map dPhi. It now follows from Theorem 6.1...' But Lemma 6.3 proves only continuity of dPhi; it does not establish injectivity or surjectivity. The proof of Theorem 6.1 uses (dPhi)^* being an isomorphism on Cech cohomology to pull back a nonzero class and contradict Phi(C(K)) cap B_{R0} = empty. If dPhi is merely continuous, (dPhi)^* need not be injective, so the contradiction fails. Consequently the uniform quasi-surjectivity that underpins the lower bound in Section 8.1 is not established as written. This is likely fixable, e.g. by proving the constructed Phi is a quasi-isometry and using the standard boundary homeomorphism, or by stating a weaker","section":"§7, Step 3; Theorem 6.1"},{"comment":"The proof of Theorem 1.4 says: 'Let K be an aspherical simplicial complex, such that Vol(K/H)=C(H,P_H). By Proposition 4.2, there exist a G-equivariant map Phi_0: L_0 -> X_0...' This is inconsistent with the setup: K is a classifying space for H, so K carries an H-action, not necessarily a G-action. Proposition 4.2 as stated requires a free cocompact G-action on K. The resolution and bound (8.1) can only be applied with H in place of G; the map should be H-equivariant. Since Theorem 7.1 is then invoked for an H-equivariant extension, I suspect this is a typo, but as written the application of Proposition 4.2 is unjustified and should be corrected by stating and using the H-version of Proposition 4.2.","section":"§8.1, Proposition 4.2 application"}],"minor_comments":[{"comment":"In the displayed inequality after (4.3), there are typos: d(\\tilde{\\Psi}((e_i)_-), \\tilde{\\Phi}((e_i)_+)) should be d(\\tilde{\\Psi}((e_i)_-), \\tilde{\\Psi}((e_i)_+)); and several parentheses are misplaced, e.g. d(\\Psi(e_i)_-), \\Psi((e_i)_+)) should be d(\\Psi((e_i)_-), \\Psi((e_i)_+)). These do not affect the argument but should be corrected.","section":"§4, Claim 4.4"},{"comment":"The displayed implication contains a stray LaTeX control sequence `/Leftr⫯g⊸tl⫯ne⇒`; it should be a simple implication arrow.","section":"§7, Equation (7.1)"},{"comment":"The arrow in the diagram `i /leftr⫯g⊸tl⫯ne →H^k_c(C(X))` is malformed and should be typeset as a standard arrow.","section":"§6, Theorem 6.1 proof"},{"comment":"The proof says 'the action G↷C(X) is free' and concludes track stabilizers are trivial. Proposition 4.2 is stated without a torsion-freeness assumption. If the proposition is intended in that generality, either add torsion-free as a hypothesis or replace 'trivial' by 'finite' (which suffices for the argument).","section":"§3, Lemma 3.4(T3)"},{"comment":"In the proof, 'local compactness of C(x)' should be 'local compactness of C(X)'.","section":"§4, Claim 4.3"}],"recommendation":"major_revision","confidential_remarks":"The Section 7 homeomorphism gap is the main obstacle; I would send the paper back for revision rather than reject it, because the strategy is plausible and the missing step is identifiable. The use of [18] is a legitimate application of prior machinery, not circular. Please ensure the H-version of Proposition 4.2 is stated explicitly in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result, but the proof as written has at least one genuine gap that needs fixing before it can be trusted. The main theorem — finite index rigidity for torsion-free relatively hyperbolic groups with NRH peripherals — is a natural extension of Lazarovich’s hyperbolic rigidity, and the applications to pinched negative curvature, limit groups, and free-by-cyclic groups are substantial. The new technical core is a uniform lower bound on the complexity of relative classifying spaces over all finite-index subgroups, combining weighted patterns from [18] with Manning–Wang’s cohomological results. That is the right strategy and the result is likely true.\n\nThe serious issue is in Section 7. Theorem 6.1 requires the map Φ to extend to a homeomorphism of Bowditch boundaries. The proof only cites Lemma 6.3, which gives a continuous extension. Continuity does not imply the boundary map is bijective, and without bijectivity the cohomology pushforward in Theorem 6.1 can collapse the nontrivial class, so the uniform quasi-surjectivity conclusion does not follow. This is a real gap in the written argument.\n\nThat said, it is probably repairable. An arbitrary H-equivariant vertex map between the universal covers of finite classifying spaces for H and G is a quasi-isometry (because H has finite index in G), and the extension constructed in §7 should be a quasi-isometry of cusped spaces, which would induce the required boundary homeomorphism. The authors just don’t say that, and a referee cannot be expected to supply it silently.\n\nThere is also a smaller equivariance typo in §8.1, where Proposition 4.2 is applied to a complex with a G-action when the natural action is by H. The argument likely goes through with H substituted throughout, but as written it is inconsistent.\n\nThe paper relies on [18] and [20] for heavy machinery, which is fine. The gaps are in the new parts. I would not desk-reject this; the main idea is credible and the missing pieces are fixable. But it needs a serious referee who will either verify the boundary homeomorphism or demand the missing argument. If repaired, this is a strong paper. For a reading group, I’d say maybe — the ideas are worth discussing despite the incomplete state. I’d cite it once the gaps are patched.\n\nRecommendation: send to peer review, with the boundary-homeomorphism issue as the main point to resolve.","headline":"A significant finite-index rigidity theorem for relatively hyperbolic groups, with the right ideas, but the written proof has a genuine gap around the boundary homeomorphism that needs patching before it is airtight.","tokens_in":19776,"tokens_out":12750,"would_cite":true,"duration_ms":117786,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F67","20F65","20J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that torsion-free relatively hyperbolic groups with non-relatively-hyperbolic peripheral subgroups are finite index rigid: any two isomorphic finite-index subgroups have the same index.","keywords":["finite index rigidity","relatively hyperbolic groups","relative classifying space","complexity","Bowditch boundary","weighted singular patterns","relative cohomology","peripheral subgroups"],"falsifier":"Find a group satisfying the hypotheses of Theorem 1.1, for instance a torsion-free group hyperbolic relative to a non-virtually-cyclic nilpotent subgroup, with two finite-index subgroups H and H' that are isomorphic but have [G:H] ≠ [G:H']. The theorem predicts no such pair exists; a more localized check is whether the boundary extension built in Section 7 can fail to be bijective, which would invalidate the application of Theorem 6.1.","tokens_in":18798,"feed_emoji":"🧩","tokens_out":5270,"duration_ms":53668,"temperature":0.7,"pith_summary":"The paper proves a rigidity theorem for a large class of groups: if a torsion-free group G is hyperbolic relative to proper subgroups that are themselves not relatively hyperbolic, then G is finite index rigid—isomorphic finite-index subgroups must have equal index. It also proves a more general statement: whenever an isomorphism between finite-index subgroups preserves the induced peripheral structure, the indices are equal, with no extra hypothesis on the peripheral subgroups. The proof introduces a complexity invariant for a group pair, namely the minimal number of cells in a simplicial relative classifying space, and shows this complexity grows linearly with the index of a finite-index subgroup. This linear growth is then converted, via a limit-inferior multiplicative invariant, into equality of indices for isomorphic subgroup pairs.","feed_headline":"Index rigidity extends to relatively hyperbolic groups","feed_subtitle":"Proof: the cell-complexity of any finite-index subgroup grows linearly with its index.","key_machinery":"The key object is the complexity C(G,P): the minimal number of cells in a simplicial relative classifying space for a group pair (G,P), where the peripheral subgroups P are represented by subcomplexes that are classifying spaces for them. The proof shows that for a torsion-free one-ended relatively hyperbolic group, the complexity of a finite-index subgroup H with its induced peripheral structure grows at least linearly with [G:H] and at most linearly via covers of a minimal complex. The lower bound is obtained by resolving a globally stable bicombing on a combinatorial cusped hyperbolic graph into a weighted singular pattern on the 2-skeleton of the classifying complex, bounding the total w","core_discovery":"The central claim is Theorem 1.1: if G is a torsion-free group not isomorphic to Z, hyperbolic relative to a finite collection of type-F and non-relatively-hyperbolic proper subgroups, then G is finite index rigid. The proof proceeds by first establishing Theorem 1.5, which says that if two finite-index subgroups H and H' of a torsion-free relatively hyperbolic group G admit an isomorphism preserving their induced peripheral structures, then [G:H] = [G:H']. The paper then shows that, under the hypotheses of Theorem 1.1, every isomorphism between finite-index subgroups automatically preserves peripheral structures: peripheral subgroups are quasiconvex, undistorted, and non-relatively-hyperbol","pith_inferences":["The linear-growth mechanism may extend beyond relative hyperbolicity: any setup with a hyperbolic cusped space admitting a globally stable bicombing and a cohomological boundary-control theorem could support a similar complexity-vs-index argument.","The complexity invariant might be computable in explicit examples, potentially giving effective bounds on indices and an algorithmic way to detect when isomorphic finite-index subgroups cannot have different indices.","The peripheral-structure-preserving theorem suggests a relative analogue of Mostow-style rigidity: in many relatively hyperbolic groups, isomorphism of finite-index subgroups may automatically preserve peripheral structures whenever the peripherals are 'large enough' in a coarse-geometric sense.","The paper's approach could also yield finite index rigidity for groups that are only virtually torsion-free, by passing to a torsion-free finite-index subgroup and tracking index changes."],"forward_implications":["Fundamental groups of complete finite-volume manifolds of pinched negative curvature are finite index rigid.","Non-abelian limit groups are finite index rigid.","Torsion-free groups hyperbolic relative to nilpotent subgroups are finite index rigid.","Free-by-cyclic groups with an exponentially growing automorphism are finite index rigid.","For any torsion-free relatively hyperbolic group, an isomorphism between finite-index subgroups that preserves the induced peripheral structure forces equal indices."],"supporting_citations":[{"why":"Supplies the weighted singular pattern machinery and the earlier hyperbolic-group finite-index rigidity result that this paper extends.","marker":"[18]"},{"why":"Provides the combinatorial cusped space construction and the globally stable bicombing for relatively hyperbolic groups used throughout the proof.","marker":"[15]"},{"why":"Defines the cylindrical cusped space, proves the isomorphism between relative group cohomology and Bowditch boundary cohomology, and supplies the uniform quasi-surjectivity theorem (Theorem 6.1).","marker":"[20]"},{"why":"Shows that torsion-free relatively hyperbolic groups with type-F peripheral subgroups admit finite relative classifying spaces, making the complexity invariant finite.","marker":"[7]"},{"why":"Establishes the existence of globally stable bicombings for hyperbolic groups, a foundational input for the bicombing used on cusped spaces.","marker":"[21]"},{"why":"Introduces the limit-inferior multiplicative invariant that converts linear complexity bounds into equality of indices for isomorphic subgroup pairs.","marker":"[26]"},{"why":"Proves finite index rigidity for multi-ended groups, allowing the proof to reduce to the one-ended case.","marker":"[27]"},{"why":"Provides the quasiconvexity, undistortedness, and maximality properties of peripheral subgroups used to show that arbitrary isomorphisms preserve peripheral structure.","marker":"[24]"},{"why":"Shows that non-virtually-cyclic nilpotent groups are not relatively hyperbolic, supplying the key examples of NRH peripheral subgroups.","marker":"[11]"}],"fun_headline_variants":["Relatively hyperbolic groups are finite-index rigid","Isomorphic finite-index subgroups of relatively hyperbolic groups share index","Finite-index rigidity proven for relatively hyperbolic groups","Isomorphic subgroups get equal index in relatively hyperbolic groups","Relatively hyperbolic groups: isomorphism forces equal index"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof's lower bound on complexity assumes that the continuous extension of the map Phi to Bowditch boundaries constructed in Section 7 is a homeomorphism; the text only proves continuity before invoking the theorem that needs bijectivity.","fun_headline_variants_meta":{"raw":{"variants":["Relatively hyperbolic groups are finite-index rigid","Isomorphic finite-index subgroups of relatively hyperbolic groups share index","Finite-index rigidity proven for relatively hyperbolic groups","Isomorphic subgroups get equal index in relatively hyperbolic groups","Relatively hyperbolic groups: isomorphism forces equal index"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00078,"raw_usage":{"total_tokens":3230,"prompt_tokens":637,"completion_tokens":2593,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":381,"completion_tokens_details":{"reasoning_tokens":2530}},"tokens_in":381,"tokens_out":2593,"duration_ms":15747,"temperature":1.0,"reasoning_tokens":2530,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:14:46.625695+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a group satisfying the hypotheses of Theorem 1.1, for instance a torsion-free group hyperbolic relative to a non-virtually-cyclic nilpotent subgroup, with two finite-index subgroups H and H' that are isomorphic but have [G:H] ≠ [G:H']. The theorem predicts no such pair exists; a more localized check is whether the boundary extension built in Section 7 can fail to be bijective, which would invalidate the application of Theorem 6.1.","supporting_citations":[{"cited_title":"Finite index rigidity of hyperbolic groups","cited_arxiv_id":"2302.04484","evidence_quote":"Supplies the weighted singular pattern machinery and the earlier hyperbolic-group finite-index rigidity result that this paper extends."},{"cited_title":"Dehn filling in relatively hyperbolic groups","cited_arxiv_id":null,"evidence_quote":"Provides the combinatorial cusped space construction and the globally stable bicombing for relatively hyperbolic groups used throughout the proof."},{"cited_title":"Cohomology and the Bowditch boundary","cited_arxiv_id":null,"evidence_quote":"Defines the cylindrical cusped space, proves the isomorphism between relative group cohomology and Bowditch boundary cohomology, and supplies the uniform quasi-surjectivity theorem (Theorem 6.1)."},{"cited_title":"Classifying spaces and boundaries for relatively hyper- bolic groups","cited_arxiv_id":null,"evidence_quote":"Shows that torsion-free relatively hyperbolic groups with type-F peripheral subgroups admit finite relative classifying spaces, making the complexity invariant finite."},{"cited_title":"Straightening and bounded cohomology of hyperbolic groups","cited_arxiv_id":null,"evidence_quote":"Establishes the existence of globally stable bicombings for hyperbolic groups, a foundational input for the bicombing used on cusped spaces."},{"cited_title":"Volumes of Discrete Groups and Topological Complexity of Homology Spheres","cited_arxiv_id":"dg-ga/9506010","evidence_quote":"Introduces the limit-inferior multiplicative invariant that converts linear complexity bounds into equality of indices for isomorphic subgroup pairs."},{"cited_title":"Complexity volumes of splittable groups","cited_arxiv_id":null,"evidence_quote":"Proves finite index rigidity for multi-ended groups, allowing the proof to reduce to the one-ended case."},{"cited_title":"Relatively Hyperbolic Groups: Intrinsic Geometry, Alge- braic Properties, and Algorithmic Problems: Intrinsic Geometry, Algebraic Properties, and Algorithmic Problems , volume 843","cited_arxiv_id":null,"evidence_quote":"Provides the quasiconvexity, undistortedness, and maximality properties of peripheral subgroups used to show that arbitrary isomorphisms preserve peripheral structure."},{"cited_title":"Tree-graded spaces and asymptotic cones of groups","cited_arxiv_id":null,"evidence_quote":"Shows that non-virtually-cyclic nilpotent groups are not relatively hyperbolic, supplying the key examples of NRH peripheral subgroups."}],"review_version":1}