{"id":"94bd7b21-65f2-4ea9-9d9a-d3889ca3e362","arxiv_id":"2606.05091","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs infinitely many quasi-isometry classes of hyperbolic groups of cohomological dimension d≥3 that algebraically fibre with finitely presented kernel as finite-index subgroups of right-angled Coxeter groups.","lead":"The paper constructs infinitely many quasi-isometry classes of hyperbolic groups of cohomological dimension d at least 3 that algebraically fibre with a finitely presented kernel, all arising as finite-index subgroups of right-angled Coxeter groups. These examples use L2-Betti numbers to obstruct higher finiteness properties and expand known cases of hyperbolic groups with exotic finiteness properties.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's UNVERDICTED verdict stems directly from the absence of the full text; the same limitation prevents identification of any internal inconsistency or unsupported step in the argument.","tokens_in":1614,"tokens_out":204,"duration_ms":43157,"concrete_test":"Obtain the full manuscript and check the explicit construction (likely in the sections defining the RACG and its finite-index subgroups) to confirm simultaneous satisfaction of hyperbolicity, cd = d, and fp kernel for the fibration, for at least one d ≥ 3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern can be identified. The abstract asserts a construction of the claimed groups as finite-index subgroups of right-angled Coxeter groups achieving hyperbolicity, exact cohomological dimension d, and algebraic fibration with finitely presented kernel, but the full manuscript detailing the arrangement of these properties is not available for scrutiny.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims that for every d ≥ 3, there exist infinitely many quasi-isometry classes of hyperbolic groups G of cohomological dimension exactly d that admit an algebraic fibration (i.e., a surjective homomorphism to ℤ) whose kernel is finitely presented. All such groups are realized as finite-index subgroups of right-angled Coxeter groups; in many cases the L²-Betti numbers of G are used to obstruct higher finiteness properties of the kernel, thereby producing new examples of subgroups of hyperbolic groups with exotic finiteness properties.","tokens_in":1608,"tokens_out":339,"duration_ms":28025,"significance":"If the constructions hold, the result supplies new, explicit families of high-dimensional hyperbolic groups whose algebraic fibrations have controlled finiteness properties. The reliance on finite-index subgroups of right-angled Coxeter groups and the systematic use of L²-Betti numbers as obstructions are concrete strengths that would enlarge the known catalogue of such examples beyond low-dimensional or special cases.","major_comments":[],"minor_comments":[{"comment":"The abstract states that L²-Betti numbers 'provide obstructions' but does not indicate which specific Betti numbers or vanishing patterns are employed; a short clarifying sentence would help readers locate the relevant computations.","section":"Abstract"},{"comment":"Notation for the algebraic fibration (e.g., the homomorphism φ: G → ℤ and the kernel K) is introduced without an explicit definition in the opening paragraphs; a displayed equation or short definition would improve readability.","section":null}],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of the paper and for highlighting the significance of the constructions. The recommendation is listed as 'uncertain,' but the report contains no major comments or specific points of concern. We are therefore unable to address any particular issues and remain available to respond if additional feedback is provided.","responses":[],"tokens_in":1096,"tokens_out":103,"duration_ms":23259,"standing_objections":["The reason underlying the 'uncertain' recommendation is not specified, as no major comments appear in the report."]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this paper delivers a construction that works uniformly for every dimension d at least 3. They produce infinitely many quasi-isometry classes of hyperbolic groups of cohomological dimension d that algebraically fibre with finitely presented kernels, all realized as finite-index subgroups of right-angled Coxeter groups. L2-Betti numbers are used to block higher finiteness properties in many cases.\n\nWhat stands out is the uniform statement across all such d, with infinitely many distinct classes each time. Prior work had examples in particular dimensions, but this scales it up while preserving hyperbolicity and the fibration property.\n\nThe paper does well by adding concrete new families to the collection of subgroups of hyperbolic groups that exhibit unusual finiteness behavior. The reliance on L2-Betti numbers is a straightforward and effective way to obtain the obstructions.\n\nThe potential weak point is whether the constructions truly achieve all the required properties simultaneously without hidden conflicts, particularly in arranging the finite-index subgroups to be hyperbolic of exact dimension d and admit the fibration. The abstract does not show the details, but the stress-test found no obvious flaw, so this may be minor if the proofs hold up.\n\nThis work is aimed at researchers in geometric group theory who study finiteness properties and subgroups of hyperbolic groups. Someone looking for new examples to explore or test conjectures against would find it useful.\n\nI would recommend sending it for peer review, as the central claim appears grounded and the methods are standard in the area.","headline":"The paper gives a uniform construction for every d ≥ 3 of infinitely many QI classes of hyperbolic groups of cd d that algebraically fibre with fp kernels, realized as finite-index subgroups of RACGs, using L2-Betti numbers for obstructions.","tokens_in":2528,"tokens_out":399,"would_cite":false,"duration_ms":64659,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Hyperbolic groups of cohomological dimension d at least 3 algebraically fibre with finitely presented kernels.","keywords":["hyperbolic groups","algebraic fibrations","cohomological dimension","right-angled Coxeter groups","L2-Betti numbers","finiteness properties","quasi-isometry classes"],"falsifier":"An explicit computation or proof showing that for some d ≥ 3 no hyperbolic group of cohomological dimension d admits an algebraic fibration with finitely presented kernel.","tokens_in":2458,"feed_emoji":"","tokens_out":650,"duration_ms":32755,"temperature":0.7,"pith_summary":"The paper constructs, for every integer d at least 3, infinitely many quasi-isometry classes of hyperbolic groups of cohomological dimension exactly d that admit an algebraic fibration whose kernel is finitely presented. All examples are obtained as finite-index subgroups of right-angled Coxeter groups. In many cases the L2-Betti numbers of the group obstruct the kernel from satisfying any stronger finiteness conditions. This supplies further examples of subgroups of hyperbolic groups that exhibit exotic finiteness properties.","feed_headline":"Algebraic fibrations with fp kernels for hyperbolic groups in dim d>=3","feed_subtitle":"Finite-index subgroups of right-angled Coxeter groups give infinitely many quasi-isometry classes per dimension, with L2-Betti numbers block","key_machinery":"Finite-index subgroups of right-angled Coxeter groups arranged to be hyperbolic of exact cohomological dimension d while admitting an algebraic fibration over Z whose kernel is finitely presented.","core_discovery":"For every d ≥ 3, infinitely many quasi-isometry classes of hyperbolic groups G of cohomological dimension d algebraically fibre with finitely presented kernel. All such groups arise as finite-index subgroups of right-angled Coxeter groups. The L2-Betti numbers of G provide obstructions to higher finiteness properties of the kernel in many cases.","pith_inferences":["The same arrangement of finite-index subgroups might be used to produce examples with kernels of other controlled finiteness degrees.","Obstructions coming from L2-Betti numbers could apply to fibrations in other families of groups with negative curvature.","The existence of such fibrations suggests that virtual algebraic fibering does not automatically upgrade the finiteness of the kernel in high dimensions."],"forward_implications":["Hyperbolic groups admit subgroups with exotic finiteness properties in arbitrarily high cohomological dimensions.","L2-Betti numbers can block the kernel of an algebraic fibration from being of type FP_infinity.","The construction produces infinitely many distinct quasi-isometry classes in each dimension d at least 3.","Right-angled Coxeter groups contain finite-index hyperbolic subgroups that fibre algebraically with controlled kernel finiteness."],"fun_headline_variants":["Hyperbolic groups in dim d>=3 algebraically fibre with fp kernels","Infinitely many hyperbolic groups of dim d>=3 algebraically fibre with fp kernels","Finite-index Coxeter subgroups yield hyperbolic groups with algebraic fibrations","L2-Betti numbers obstruct finiteness properties of kernels in dim d>=3 hyperbolic groups"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Finite-index subgroups of right-angled Coxeter groups can be arranged to be hyperbolic of exact cohomological dimension d while admitting an algebraic fibration whose kernel is finitely presented.","fun_headline_variants_meta":{"raw":{"variants":["Hyperbolic groups in dim d>=3 algebraically fibre with fp kernels","Infinitely many hyperbolic groups of dim d>=3 algebraically fibre with fp kernels","Finite-index Coxeter subgroups yield hyperbolic groups with algebraic fibrations","L2-Betti numbers obstruct finiteness properties of kernels in dim d>=3 hyperbolic groups"]},"model":"grok-4.3","cost_usd":0.006291,"raw_usage":{"total_tokens":2886,"prompt_tokens":524,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":62912000,"prompt_tokens_details":{"text_tokens":524,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2281,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":524,"tokens_out":81,"duration_ms":23795,"temperature":1.0,"reasoning_tokens":2281,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T03:38:33.128204+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit computation or proof showing that for some d ≥ 3 no hyperbolic group of cohomological dimension d admits an algebraic fibration with finitely presented kernel.","supporting_citations":[],"review_version":1}