{"id":"75017940-8c37-4df0-a52b-dba00cb4f754","arxiv_id":"1908.07946","paper_version":7,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Hyperbolic TDLC-groups of rational discrete cohomological dimension at most 2 have every compactly presented closed subgroup hyperbolic.","lead":"This paper proves that, in totally disconnected locally compact groups, hyperbolicity is passed down to compactly presented closed subgroups whenever the ambient group has rational cohomological dimension at most two. It also produces new examples from automorphism groups of hyperbolic buildings and from small cancellation quotients of profinite groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.1 as written does not establish the commuting diagram: defining δ_n = π∂_n only commutes if ∂_n(Ω_n) lies in ker δ_{n-1}, which the construction does not ensure.","rationale":"The reader identified Proposition 3.7, the Rips-complex carry-over, as the weakest assumption. That carry-over is standard and likely sound: Cayley-Abels graphs are locally finite, vertex-transitive hyperbolic graphs, and the Rips complex for sufficiently large parameter is finite-dimensional, locally finite, contractible, and has compact open cell stabilizers with finitely many orbits. The more serious problem is internal to the proof of Lemma 5.1, which is load-bearing for Theorem 1.5 and hence for the main theorem. The inductive construction defines δ_n through the retraction π, but the commuting square required in the lemma is only guaranteed if ∂_n sends every element of the constructed Ω_n into ker δ_{n-1}. The text never proves this, and the free choice of preimages makes it visibly false in an elementary example. Because Theorem 1.5 is the mechanism that transfers the weak linear isoperimetric inequality from G to H, a gap here leaves the central claim without proof as written. The result may still be true and the lemma repairable by a more careful relative-resolution argument, so a conditional verdict is appropriate rather than rejection. The paper contains substantial supporting material, including the characterization in Theorem 1.4 and the small-cancellation examples, but the subgroup-inheritance step is not established by the present proof.","tokens_in":21940,"tokens_out":37678,"duration_ms":331576,"concrete_test":"Re-run the inductive construction of Lemma 5.1 for the explicit inclusion H = Z in G = F_2 = <a,b>. Use the G-resolution Q[G]^2 → Q[G] with ∂_1(e_a) = a-1 and ∂_1(e_b) = b-1, and the H-resolution from the induction. For the generator x = a-1 of ker δ_0, the proof permits the preimage y = e_a + e_b, since π∂_1(y) = π(a-1 + b-1) = a-1. The constructed Ω_1 then contains e_b, and δ_1(e_b) = π∂_1(e_b) = π(b-1) = 0, while ∂_1(e_b) = b-1 ≠ 0 in Q[G]. Consequently the square does not commute. Check whether the lemma's proof can be modified to force a choice of preimages with ∂_n(α) ∈ ker δ_{n-1} for all components α; if not, Lemma 5.1 as stated is unproved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.5, the core of Theorem 1.1, rests on Lemma 5.1. In the inductive step of Lemma 5.1, the proof chooses preimages y_i of generators x_i of ker δ_{n-1} under the map Q[Σ_n] → ker ∂_{n-1} → ker δ_{n-1}, defines Ω_n as the H-subset generated by the components α_{ij} of the y_i, and then sets δ_n = π∂_n on Q[Ω_n]. For the displayed diagram to commute, one needs ∂_n(α) = ι_{n-1}δ_n(α) = π∂_n(α) for every basis element α ∈ Ω_n, equivalently ∂_n(α) ∈ ker δ_{n-1} for every α. The proof controls only the projected images π∂_n(y_i) = x_i; it does not show that the individual components α_{ij} satisfy ∂_n(α_{ij}) ∈ ker δ_{n-1}. Indeed the permitted preimages are not unique, and a non-minimal choice can introduce components whose boundary lies outside ker δ_{n-1}. If the square fails, the vertical inclusion does not induce a well-defined map ker δ_n → ker ∂_n, so condition (5) and the subsequent conclusion that ker δ_n is a summand of ker ∂_n are unsupported. This is not a cosmetic gap: Remark 5.2 explicitly notes that a mapping-cylinder argument works only for open subgroups, so the closed-subgroup case is exactly what Lemma 5.1 is designed to handle. Without a valid proof of Lemma 5.1, Theorem 1.5, and hence Theorem 1.1, lack a demonstrated proof.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies large-scale geometric properties of totally disconnected locally compact (TDLC) groups. It introduces a weak n-dimensional linear isoperimetric inequality and proves that a compactly generated TDLC-group is hyperbolic if and only if it is compactly presented and satisfies the weak linear isoperimetric inequality (Theorem 1.4). The main theorem (Theorem 1.1) states that every compactly presented closed subgroup of a hyperbolic TDLC-group with rational discrete cohomological dimension at most 2 is hyperbolic. The proof combines Theorem 1.4 with Theorem 1.5, an inheritance result for the weak n-dimensional linear isoperimetric inequality under passage to closed subgroups of type FP_{n+1}. Applications are given to automorphism groups of locally finite 2-dimensional CAT(-1) complexes, including Bourdon buildings, and to small cancellation quotients of amalgamated products of profinite groups. The paper also discusses a variant with asymptotic dimension in place of cohomological dimension, proving it for discrete groups and sketching the TDLC case.","tokens_in":22237,"tokens_out":12905,"duration_ms":202915,"significance":"If the proof is completed, this would be a substantial generalization of Gersten's theorem on subgroups of hyperbolic groups in dimension 2, extending results of Arora and Martinez-Pedroza to the TDLC setting. The homological characterization of hyperbolicity (Theorem 1.4) is a useful new tool, and the applications to automorphism groups of buildings are natural and interesting. The paper is generally clearly written and the overall strategy is transparent. However, the proof of the key technical lemma (Lemma 5.1) has a commutativity gap that affects Theorem 1.5 and hence the main theorem; this must be repaired before the results can be accepted.","major_comments":[{"comment":"The proof of Lemma 5.1 does not establish the commutativity of the displayed diagram. The map δ_n is defined as π∂_n on Q[Ω_n]. For the square to commute, one needs ∂_n(α) = ι_{n-1}δ_n(α) = ι_{n-1}π∂_n(α) for every basis element α ∈ Ω_n, equivalently ∂_n(α) ∈ ker δ_{n-1}. The construction only ensures that π∂_n(y_i) = x_i for the chosen preimages y_i of the generators x_i of ker δ_{n-1}; it does not control the individual summands α_{ij} of y_i, whose boundaries may lie outside ker δ_{n-1}. Consequently the induced map ker δ_n → ker ∂_n may not be well-defined, condition (5) (projectivity of coker(ker δ_n → ker ∂_n)) is not justified, and the proof of Theorem 1.5 collapses. Since Theorem 1.5 is the engine behind Theorem 1.1, this is a load-bearing gap. Remark 5.2 makes clear that the open-subgroup case is not sufficient, so the closed-subgroup argument must be supplied.","section":"Section 5, Lemma 5.1"},{"comment":"Proposition 3.7 states that a hyperbolic TDLC-group acts on a finite-dimensional contractible locally finite simplicial complex with compact open cell stabilizers and finitely many cell orbits, citing that the proof of [10, III.Γ Theorem 3.21] carries over to Cayley-Abels graphs. Since this proposition is used to obtain compact presentability and finite type FP_∞ in Theorem 1.4 and Theorem 1.1, and since the action on a Cayley-Abels graph has non-trivial (compact open) vertex stabilizers, the carry-over is not entirely routine. Please provide a detailed argument (or a precise reference) that the Rips complex on a Cayley-Abels graph is locally finite, finite-dimensional, contractible for a large parameter, and has finitely many G-orbits of cells with compact open stabilizers.","section":"Section 3.3, Proposition 3.7"}],"minor_comments":[{"comment":"There is a typo: 'posibble' should be 'possible'.","section":"Section 1, paragraph 1"},{"comment":"After defining Ω_n, the sentence 'We get an induced map...' should explicitly state that the commutativity of the diagram is being asserted and then prove it; currently the commutativity is assumed rather than demonstrated.","section":"Section 5, proof of Lemma 5.1"},{"comment":"The proof should explicitly address the cases cd_Q(G) = 0 and cd_Q(G) = 1; for instance, cd_Q(G) = 0 makes G profinite, and cd_Q(G) = 1 can be handled via the n = 0 case of Theorem 1.5 or a separate argument.","section":"Proof of Theorem 1.1"},{"comment":"There is a typo: 'it follows tha that' should be 'it follows that'.","section":"Section 4.6, proof of Proposition 4.7"},{"comment":"The title contains spacing errors: 'TOT ALL Y' and 'COMP ACT' should be 'TOTALLY' and 'COMPACT'.","section":"Title page"}],"recommendation":"major_revision","confidential_remarks":"The gap in Lemma 5.1 is serious and directly affects the proof of the main theorem. The paper is otherwise well organized and the results are attractive, so I would be willing to referee a revised version that repairs this argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper does a lot of things well. It sets up a clean cohomological framework for TDLC groups, proves a characterization of hyperbolicity via weak linear isoperimetric inequalities (Theorem 1.4), and uses it to attack a natural subgroup inheritance problem. The applications to automorphism groups of CAT(-1) buildings and the small cancellation examples are genuinely attractive, and the authors are honest about what is new and what is borrowed, including Remark 5.2 on why open subgroups are easier.\n\nThe problem is Lemma 5.1, which is the load-bearing step for Theorem 1.5 and hence for the main Theorem 1.1. The construction of the H-resolution is incomplete. The authors pick preimages y_i in Q[Σ_n] of generators of ker(δ_{n−1}), then let Ω_n be generated by the components α_ij appearing in those y_i. But to make the square in condition (4) commute, you need ∂_n(α_ij) to lie in Q[Ω_{n−1}] for each component α_ij, or at least in ker(δ_{n−1}) with the retraction behaving correctly. The proof only controls the projected images π∂_n(y_i) = x_i; it says nothing about the individual components. If a component has boundary outside Q[Ω_{n−1}], the square fails, and then the map ker(δ_n) → ker(∂_n) is not even defined. The argument in the diagram following the lemma assumes this map exists, so condition (5) is not established.\n\nIs this fatal? For the proof as written, yes. Theorem 1.5 relies entirely on Lemma 5.1, and Theorem 1.1 relies on Theorem 1.5. I can imagine a fix—for instance, choosing the preimages more carefully so that each boundary lands in the relevant submodule, or enlarging Ω_{n−1} during the induction—but the text doesn't provide it. This is not a cosmetic omission; it is exactly the point where the closed-subgroup case differs from the open-subgroup case, as the authors themselves stress in Remark 5.2.\n\nMinor issues: the sketch of Lemma 4.8 is terse, and Proposition 4.1's proof has a slight misstatement about the matrix entries, but neither affects the central argument unless the chain collapses elsewhere.\n\nVerdict: the paper deserves a serious referee and a chance for revision, but I would not accept it until the proof of Lemma 5.1 is repaired. I would not cite the main theorem in its current form, though I might cite the characterization in Theorem 1.4 if that part is independently solid.","headline":"A strong framework and a likely-true main theorem, but the proof of Lemma 5.1 has a real commutativity gap that leaves Theorem 1.1 unproven as written.","tokens_in":22849,"tokens_out":5806,"would_cite":false,"duration_ms":105638,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F67","22D05","20J05","20E06","20F65"],"pacs":[],"model":"deepseek-v4-flash","headline":"In hyperbolic TDLC-groups of rational discrete cohomological dimension at most 2, every compactly presented closed subgroup is hyperbolic.","keywords":["hyperbolic groups","totally disconnected locally compact groups","homological finiteness","cohomological dimension 2","compactly presented","weak linear isoperimetric inequality","small cancellation","CAT(-1) buildings"],"falsifier":"The central claim predicts a norm inequality: in a partial proper permutation resolution for a hyperbolic TDLC-group $G$, every element of $\\ker(\\delta_1)$ is fillable with cost linear in its $\\ell^1$-norm, and the same must hold for any compactly presented closed subgroup. One could try to construct a pair $(G,H)$ with $cd_{\\mathbb{Q}}(G) \\le 2$ and $H$ compactly presented but with no uniform filling constant; finding one would refute Theorem 1.1.","tokens_in":21686,"feed_emoji":"📐","tokens_out":10256,"duration_ms":81540,"temperature":0.7,"pith_summary":"This paper proves that hyperbolicity is inherited by compactly presented closed subgroups within a large class of non-discrete locally compact groups: totally disconnected locally compact (TDLC) groups whose rational discrete cohomological dimension is at most 2. The proof characterizes hyperbolic TDLC-groups by a homological isoperimetric inequality and then shows that this inequality transfers from a group to certain closed subgroups. A reader should care because this extends a classical theorem for discrete hyperbolic groups to the TDLC setting, where compact open subgroups replace finite subgroups. The result also gives concrete consequences, including that every compactly presented closed subgroup of the automorphism group of a negatively curved locally finite 2-dimensional building is hyperbolic whenever that automorphism group acts with finitely many orbits.","feed_headline":"Subgroups of low-dimensional hyperbolic TDLC groups stay hyperbolic","feed_subtitle":"Compactly presented closed subgroups inherit hyperbolicity, via a proof through homological isoperimetric inequalities.","key_machinery":"The load-bearing notion is the weak $n$-dimensional linear isoperimetric inequality for a TDLC-group $G$ of type $FP_{n+1}$. Taking a finite-type proper permutation resolution $\\mathbb{Q}[\\Omega_{n+1}] \\to \\mathbb{Q}[\\Omega_n] \\to \\cdots \\to \\mathbb{Q}[\\Omega_0] \\to \\mathbb{Q} \\to 0$, the kernel of $\\delta_n$ must be undistorted in $\\mathbb{Q}[\\Omega_n]$, meaning its filling norm is equivalent to the inherited $\\ell^1$-norm. For $n=1$ this is the weak linear isoperimetric inequality, and Theorem 1.4 identifies it with hyperbolicity of compactly generated TDLC-groups, following a homological characterization of hyperbolic $2$-complexes. The transfer is carried by Lemma 5.1, which builds compatible partial proper permutation resolutions for $G$ and a closed subgroup $H$ so that the cokernel of the induced map on kernels is projective; Theorem 1.5 then compares filling norms through that diagram. A second load-bearing input is Proposition 3.7, which carries the Rips-complex construction from discrete hyperbolic groups to Cayley-Abels graphs, making hyperbolic TDLC-groups compactly presented and of type $FP_\\infty$.","core_discovery":"The central claim is Theorem 1.1: if $G$ is a hyperbolic TDLC-group with $cd_{\\mathbb{Q}}(G) \\le 2$, then every compactly presented closed subgroup $H$ of $G$ is hyperbolic. The proof has two main steps. Theorem 1.4 says a compactly generated TDLC-group is hyperbolic exactly when it is compactly presented and satisfies the weak linear isoperimetric inequality. Theorem 1.5 says that, inside a TDLC-group $G$ of type $FP_\\infty$ with $cd_{\\mathbb{Q}}(G) = n+1$, every closed subgroup $H$ of type $FP_{n+1}$ inherits the weak $n$-dimensional linear isoperimetric inequality. Applying Theorem 1.5 with $n=1$ and then Theorem 1.4 again yields the subgroup theorem. The paper also constructs examples: small cancellation quotients of amalgamated free products of profinite groups over open subgroups are hyperbolic TDLC-groups of rational discrete cohomological dimension at most 2, and the building automorphism group application follows.","pith_inferences":["One could test whether the hypothesis that $H$ is compactly presented can be relaxed to type $FP_2$; Question 1 leaves this open, and the proof suggests the obstruction is the projectivity of the cokernel in Lemma 5.1.","If the boundary cohomology formula $cd_{\\mathbb{Q}}G = \\dim_{\\mathbb{Q}} \\partial_\\infty G + 1$ holds for hyperbolic TDLC-groups, then the proof of Theorem 1.6 should extend and answer the asymptotic-dimension question positively for TDLC-groups.","The small-cancellation construction suggests further examples: apply the same quotients to other finite graphs of profinite groups and check whether the resulting TDLC-groups still have rational discrete cohomological dimension at most 2, which would widen the domain of the subgroup theorem."],"forward_implications":["Every compactly presented closed subgroup of a hyperbolic TDLC-group of rational discrete cohomological dimension at most 2 is itself hyperbolic.","If $X$ is a locally finite 2-dimensional simplicial $\\mathrm{CAT}(-1)$-complex and $\\mathrm{Aut}(X)$ acts with finitely many orbits on $X$, then every compactly presented closed subgroup of $\\mathrm{Aut}(X)$ is hyperbolic; right-angled hyperbolic polygon buildings provide examples.","Small cancellation quotients of amalgamated free products of profinite groups over open subgroups, under the $C'(1/12)$ condition, produce hyperbolic TDLC-groups with $cd_{\\mathbb{Q}} \\le 2$.","For discrete hyperbolic groups, the paper proves that if $\\operatorname{asdim} G \\le 2$, then every finitely presented subgroup of $G$ is hyperbolic, using $cd_{\\mathbb{Q}}G \\le \\operatorname{asdim} G$."],"supporting_citations":[{"why":"establishes the discrete rational-dimension-2 subgroup theorem that this paper extends to TDLC-groups.","marker":"[3]"},{"why":"supplies the original discrete subgroup theorem in integral dimension 2 and the isoperimetric strategy adapted here.","marker":"[22]"},{"why":"constructs rational discrete cohomology and the category of discrete modules in which type $FP_n$ and $cd_{\\mathbb{Q}}$ are defined.","marker":"[17]"},{"why":"provides the characterization of hyperbolic simply connected 2-complexes by a linear homological isoperimetric inequality, used in Theorem 1.4.","marker":"[25]"},{"why":"contains the Rips-complex theorem for hyperbolic groups that Proposition 3.7 carries over to Cayley-Abels graphs.","marker":"[10]"},{"why":"gives the boundary cohomology formula used to deduce $cd_{\\mathbb{Q}}G \\le \\operatorname{asdim} G$ for discrete hyperbolic groups.","marker":"[7]"},{"why":"gives the asymptotic dimension formula for hyperbolic spaces, also used in the proof of Theorem 1.6.","marker":"[12]"},{"why":"supplies the small cancellation theory for amalgamated free products used in Theorem 7.1.","marker":"[31]"}],"fun_headline_variants":["Low-dim hyperbolic TDLC groups: compactly presented closed subgroups are hyperbolic","Hyperbolic TDLC groups of cohomological dimension ≤2: subgroups stay hyperbolic","Subgroups of low-dim hyperbolic TDLC groups inherit hyperbolicity","Hyperbolicity passes to closed subgroups in low-dim hyperbolic TDLC groups","In hyperbolic TDLC groups with dim ≤2, closed subgroups remain hyperbolic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the fact that a standard device for discrete hyperbolic groups, building a large contractible complex from nearby vertices of a Cayley graph, works unchanged for Cayley-Abels graphs of TDLC-groups; if that carry-over failed for non-discrete groups, the proof of Theorem 1.1 would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Low-dim hyperbolic TDLC groups: compactly presented closed subgroups are hyperbolic","Hyperbolic TDLC groups of cohomological dimension ≤2: subgroups stay hyperbolic","Subgroups of low-dim hyperbolic TDLC groups inherit hyperbolicity","Hyperbolicity passes to closed subgroups in low-dim hyperbolic TDLC groups","In hyperbolic TDLC groups with dim ≤2, closed subgroups remain hyperbolic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001307,"raw_usage":{"total_tokens":5365,"prompt_tokens":1020,"completion_tokens":4345,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":636,"completion_tokens_details":{"reasoning_tokens":4248}},"tokens_in":636,"tokens_out":4345,"duration_ms":31743,"temperature":1.0,"reasoning_tokens":4248,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:55:46.926256+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The central claim predicts a norm inequality: in a partial proper permutation resolution for a hyperbolic TDLC-group $G$, every element of $\\ker(\\delta_1)$ is fillable with cost linear in its $\\ell^1$-norm, and the same must hold for any compactly presented closed subgroup. One could try to construct a pair $(G,H)$ with $cd_{\\mathbb{Q}}(G) \\le 2$ and $H$ compactly presented but with no uniform filling constant; finding one would refute Theorem 1.1.","supporting_citations":[{"cited_title":"Subgroups of word hyperbolic groups in ra- tional dimension 2","cited_arxiv_id":null,"evidence_quote":"establishes the discrete rational-dimension-2 subgroup theorem that this paper extends to TDLC-groups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the original discrete subgroup theorem in integral dimension 2 and the isoperimetric strategy adapted here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"constructs rational discrete cohomology and the category of discrete modules in which type $FP_n$ and $cd_{\\mathbb{Q}}$ are defined."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the characterization of hyperbolic simply connected 2-complexes by a linear homological isoperimetric inequality, used in Theorem 1.4."},{"cited_title":"Bridson and Andr´ e Haefliger","cited_arxiv_id":null,"evidence_quote":"contains the Rips-complex theorem for hyperbolic groups that Proposition 3.7 carries over to Cayley-Abels graphs."},{"cited_title":"The boundary of negati vely curved groups","cited_arxiv_id":null,"evidence_quote":"gives the boundary cohomology formula used to deduce $cd_{\\mathbb{Q}}G \\le \\operatorname{asdim} G$ for discrete hyperbolic groups."},{"cited_title":"Buyalo and Nina D","cited_arxiv_id":null,"evidence_quote":"gives the asymptotic dimension formula for hyperbolic spaces, also used in the proof of Theorem 1.6."}],"review_version":1}