{"id":"5ae525b9-9595-4c6b-8867-480f7aa69f6a","arxiv_id":"1908.04894","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that if a normal finite-index subgroup of G is H-accessible or AH-accessible, then G itself is H-accessible or AH-accessible.","lead":"This paper proves that the properties H-accessible and AH-accessible, which track whether a group has a richest hyperbolic geometry, are preserved when passing from a group to a finite extension. It answers an open question from the author's earlier work and gives a structural closure result for acylindrically hyperbolic groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the transfer-lemma premise is standard and the omitted Prop 1.5 largest-element step is genuinely routine.","rationale":"I read the full proofs. Proposition 1.4 is correct: the restriction H↷Γ(G,Z) is cobounded and acylindrical; the transfer lemma yields [W]∈AH(H); since [X] is largest, [W]≼[X]; the displayed inequalities give sup_{x∈X}|x|_Z<∞ and hence [Z]≼[X∪Y]. Proposition 1.5's proof that Γ(H,X) is H-equivariantly quasi-isometric to Γ(G,X∪Y) is valid, with normality of H supplying the conjugation bounds from invariance of the largest element. The final largest-element step is stated rather than proved, but it is the same transfer argument with acylindricity removed, so no new assumption is needed. The only residual uncertainty is the exact wording of the cited lemma, which is standard Svarc-Milnor material. The reader's conditional is based on a sketched step rather than an identified error; my read agrees that this step is load-bearing but finds it secure, so I would not alter the verdict.","tokens_in":5654,"tokens_out":18521,"duration_ms":174353,"concrete_test":"Verify the exact statement of [1, Lemma 3.11]: confirm that it produces a generating set W of H with H↷Γ(H,W) equivalent to any cobounded H-action on a hyperbolic space, without requiring the action to be acylindrical, and that the coarse equivalence preserves acylindricity when present. If the lemma has no hidden finite-generation or properness hypothesis, the Proposition 1.5 largest-element step follows from Proposition 1.4 mutatis mutandis and the paper is correct as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing mathematical flaw in the central argument. The key premise, use of the Svarc-Milnor transfer lemma [1, Lemma 3.11], is secure: H↷Γ(G,Z) is cobounded because G=HY with Y finite and each y∈Y has finite word length; if the original action is acylindrical, acylindricity passes to the subgroup and is preserved under the coarse equivalence, so the transferred structure lies in AH(H) (or in H(H) when acylindricity is not imposed). The inequalities in Proposition 1.4 correctly show sup_{x∈X}|x|_Z<∞, whence [Z]≼[X∪Y]. Proposition 1.5 gives a valid H-equivariant quasi-isometry between Γ(H,X) and Γ(G,X∪Y); the largest-element assertion is sketched, but it is exactly the Proposition 1.4 argument with acylindricity dropped, so no hidden assumption appears. The finite-index and normality hypotheses are used precisely where needed. I therefore do not see a correctness risk in the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two preservation results for finite extensions of groups. Proposition 1.4 states that if a group G contains a normal, finite-index, acylindrically hyperbolic subgroup H that is AH-accessible, then G is AH-accessible. Proposition 1.5 states that if G contains a normal, finite-index subgroup H that is H-accessible, then G is H-accessible. The proofs use the largest structure [X] in AH(H) (respectively H(H)) and a finite set Y of coset representatives, and show that [X ∪ Y] is the largest structure in the corresponding poset of G. The key technical step is the Svarc-Milnor transfer lemma [1, Lemma 3.11], which allows an arbitrary structure [Z] on G to be compared with a structure [W] on H, after which [W] is dominated by the largest element [X] of H. The paper also derives a corollary for right-angled Coxeter groups containing a right-angled Artin group as a finite-index normal subgroup, and thereby answers an open question from [1, Problem 8.10].","tokens_in":5868,"tokens_out":16144,"duration_ms":154484,"significance":"If correct, the result answers an explicit open question and gives a positive answer to a special case of the quasi-isometry invariance question for acylindrical hyperbolicity (Question 1.1), namely the case of finite-index inclusions. The proof is short, transparent, and builds directly on the framework of Abbott, Balasubramanya, and Osin [1] and on the Minasyan-Osin construction [8, Lemma 6]. The manuscript is honest about the reliance on these prior results, and the main inequalities in Proposition 1.4 are explicit and checkable. The corollary on right-angled Coxeter groups is a useful concrete application. The paper does not contain machine-checked proofs, but the mathematical steps are elementary and the cited transfer lemma is standard in this context.","major_comments":[],"minor_comments":[{"comment":"The lower bound in the quasi-isometry inequality is typeset inconsistently; it should read (1/C)d_Z(1,g) - C, not \"- C + 1 C d_Z(1,g)\" as currently printed.","section":"§3, proof of Proposition 1.4, Eq. (1)"},{"comment":"There is a spurious inverse in the displayed inequality: it should be |y^{-1}xy|_X, not |y^{-1}x^{-1}y|_X.","section":"§3, proof of Proposition 1.5, Eq. (4)"},{"comment":"The notation in the coset-representative decomposition is ambiguous and contains a missing inverse: the term \"yn−1anyn\" should be y_{n-1} a_n y_n^{-1} (with y_n = 1), and the expression y_{i-1} a_i y_i^{-1} should be written with unambiguous subscripts throughout the paragraph.","section":"§3, proof of Proposition 1.5, decomposition"},{"comment":"The claim that [X∪Y] is the largest element of H(G) is dismissed as \"almost identical\" to the proof of Proposition 1.4; since this is the central claim of the proposition, please spell out the transfer step for a general [Z]∈H(G) so the reader can verify it without re-deriving the whole argument.","section":"§3, proof of Proposition 1.5, last paragraph"},{"comment":"The sentence \"Since H has finite index in G, the action is also cobounded\" would benefit from one additional sentence: each y∈Y has finite Z-length because Z generates G, so the H-orbit of 1 is cobounded in Γ(G,Z).","section":"§3, proof of Proposition 1.4, coboundedness"},{"comment":"Please state explicitly that the membership [X∪Y]∈AH(G) is precisely the construction extracted from the proof of [8, Lemma 6], because this membership is needed to conclude that [X∪Y] is the largest element of AH(G) rather than merely an upper bound for those elements.","section":"§3, proof of Proposition 1.4, use of [8, Lemma 6]"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound in its main claims and the deficiencies are limited to presentation and to the terseness of one routine but load-bearing step. The reliance on [1] and [8] is appropriate and fully cited; the novelty is incremental but it answers an explicit open question. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The upshot: this short paper closes a specific open problem. If H is a normal finite-index subgroup of G and H is H-accessible (resp. AH-accessible), then so is G. The AH-accessible half strengthens Minasyan–Osin’s Lemma 6, which only concluded that G is acylindrically hyperbolic; the H-accessible half is new. The argument is clean and the main inequalities check out.\n\nThe proof of Prop 1.4 takes an arbitrary [Z] in AH(G), restricts the action to H, applies the Svarc–Milnor transfer lemma [1, Lemma 3.11] to get [W] in AH(H), and uses the largest [X] in AH(H) to bound sup_{x in X} |x|_Z. That is exactly the right strategy, and I don't find a gap. The transfer lemma is legitimate: finite index plus finite coset representatives gives coboundedness, and acylindricity passes to subgroups and is preserved under the equivalence.\n\nProp 1.5 repeats the first part with acylindricity dropped, establishes an H-equivariant quasi-isometry between Gamma(H,X) and Gamma(G,X union Y), and then says the largest-element claim follows “almost identically” to Prop 1.4. This is the one soft spot. It is cosmetic in the sense that I see no hidden issue—the same comparison goes through without acylindricity—but it is a genuine omission in the written proof. The paper should spell out those few lines. That is a minor revision, not a substantive objection.\n\nThe citation pattern is honest. The paper leans on the author's own framework [1] and on [8]; since [1] is published and the transfer lemma is cited precisely, self-citation is not a problem. There is no circularity: the largest element in H(H) is assumed, not derived from the conclusion.\n\nThis is not a breakthrough; it is a competent, useful piece of geometric group theory. It will be of interest to people working with hyperbolic structures, universal actions, and quasi-isometry questions. I would happily see it in the literature and would cite it in related work.\n\nRecommendation: send it to peer review. A referee should ask for the expanded Prop 1.5 and maybe a sentence on why the transferred structure lies in H(H) when acylindricity is dropped, then accept.","headline":"A short, sound paper that proves H- and AH-accessibility pass to finite extensions, answering an explicit open problem; the only real soft spot is a sketched largest-element step in Proposition 1.5.","tokens_in":6390,"tokens_out":2790,"would_cite":true,"duration_ms":25818,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F67","20E22"],"pacs":[],"model":"deepseek-v4-flash","headline":"If a normal finite-index subgroup of a group is $\\mathcal{H}$-accessible or $\\mathcal{AH}$-accessible, then the containing group is also $\\mathcal{H}$-accessible or $\\mathcal{AH}$-accessible, respectively.","keywords":["acylindrically hyperbolic","hyperbolic structures","AH-accessibility","H-accessibility","finite extensions","finite-index subgroups","largest action","geometric group theory"],"falsifier":"Exhibit a finitely generated group $G$, a normal finite-index subgroup $H$, and a hyperbolic or acylindrically hyperbolic generating set $Z$ of $G$ such that, with $[X]$ the largest structure of $H$, $\\sup_{x\\in X\\cup Y}|x|_Z=\\infty$. Since the proof would then fail at the domination step, such an example would refute the proposition.","tokens_in":5457,"feed_emoji":"📐","tokens_out":10763,"duration_ms":90220,"temperature":0.7,"pith_summary":"The paper proves that two refinements of hyperbolicity for groups are preserved when passing from a group to a finite extension. Specifically, if a normal finite-index subgroup $H$ is $\\mathcal{H}$-accessible, meaning its hyperbolic structures have a largest element, or $\\mathcal{AH}$-accessible, meaning its acylindrically hyperbolic structures have a largest element, then the containing group $G$ has the same property. In both cases the largest structure of $G$ is built explicitly as $[X \\cup Y]$, where $[X]$ is the largest structure of $H$ and $Y$ is a finite set of coset representatives. This answers an open question from the foundational study of hyperbolic structures on groups.","feed_headline":"Accessibility survives finite extensions","feed_subtitle":"If a normal finite-index subgroup has a largest hyperbolic or acylindrically hyperbolic structure, so does the overgroup.","key_machinery":"The central objects are the posets $\\mathcal{H}(G)$ and $\\mathcal{AH}(G)$: equivalence classes of generating sets whose Cayley graphs are hyperbolic, or hyperbolic with an acylindrical $G$-action, ordered by domination. The load-bearing tool is the Svarc-Milnor transfer lemma from reference [1], which converts the restricted action of a finite-index subgroup $H$ on any hyperbolic Cayley graph $\\Gamma(G,Z)$ into an equivalent action of $H$ on some $\\Gamma(H,W)$. Because $[X]$ is the largest structure on $H$, one gets $[W]\\preceq [X]$, and this yields the uniform bound $\\sup_{x\\in X}|x|_Z<\\infty$; finiteness of $Y$ then gives $[Z]\\preceq [X\\cup Y]$. In the $\\mathcal{H}$-case, normality of $H$ is used to show that the largest element of $\\mathcal{H}(H)$ is invariant under conjugation by elements of $Y$, giving the bound $|y^{-1}xy|_X\\le L$ that makes the inclusion $\\Gamma(H,X)\\to\\Gamma(G,X\\cup Y)$ a quasi-isometry.","core_discovery":"The paper's central claim is that $\\mathcal{H}$-accessibility and $\\mathcal{AH}$-accessibility are preserved under finite extensions. Concretely: if $G$ contains a normal subgroup $H$ of finite index and $H$ is $\\mathcal{AH}$-accessible, then $G$ is $\\mathcal{AH}$-accessible; if $H$ is $\\mathcal{H}$-accessible, then $G$ is $\\mathcal{H}$-accessible. The proof shows that the largest element of $\\mathcal{H}(G)$ or $\\mathcal{AH}(G)$ is $[X\\cup Y]$, where $[X]$ is the largest element of $\\mathcal{H}(H)$ or $\\mathcal{AH}(H)$ and $Y$ is a finite set of distinct coset representatives of $H$ in $G$. This structure is hyperbolic, and acylindrical in the $\\mathcal{AH}$ case, and it dominates every other hyperbolic structure on $G$. In the $\\mathcal{H}$ case, normality of $H$ is used to show that the inclusion of the Cayley graph of $H$ with respect to $X$ into the Cayley graph of $G$ with respect to $X\\cup Y$ is a quasi-isometry.","pith_inferences":["The transfer strategy suggests a route to quasi-isometric invariance of $\\mathcal{AH}$-accessibility for any two groups with a common finite-index subgroup, if an analogue of the normality-based conjugation bound can be found for non-normal subgroups.","One may expect analogous preservation results for other posets of group actions, such as relatively hyperbolic or $\\mathrm{CAT(0)}$ structures, whenever a Svarc-Milnor transfer lemma is available.","Because the proof uses automorphism-invariance of the largest structure of $H$, a natural test case for the sharpness of the theorem would be a group whose largest structure is not preserved by all automorphisms; such a group might fail to be a finite extension with the property."],"forward_implications":["If the hypotheses hold, $G$ has a largest acylindrically hyperbolic structure, so it admits a universal acylindrical action in which every generalized loxodromic element acts loxodromically.","In this finite-index situation, $\\mathcal{AH}$-accessibility, and hence acylindrical hyperbolicity, is invariant under the quasi-isometry between $H$ and $G$, giving a special-case affirmative answer to the wider quasi-isometry question.","The proof is constructive: the largest structure of $G$ is $[X\\cup Y]$, so one can build the largest action of the extension explicitly from the largest action of the subgroup.","Right-angled Coxeter groups containing a right-angled Artin group as a finite-index normal subgroup are $\\mathcal{AH}$-accessible, and their largest action is equivalent to the extension of the RAAG's extension-graph action.","The results directly answer Problem 8.10 from the paper's main reference on hyperbolic structures."],"supporting_citations":[{"why":"This reference supplies the definitions of $\\mathcal{H}(G)$ and $\\mathcal{AH}(G)$, the transfer lemma used in both proofs, and the open problem that the paper answers.","marker":"[1]"},{"why":"It proves the earlier result that $G$ is acylindrically hyperbolic under the same hypotheses, and its construction of $[X\\cup Y]$ is the starting point for the two propositions.","marker":"[8]"},{"why":"It is cited for the equivalence between the restricted action $H\\curvearrowright\\Gamma(G,Z)$ and an action $H\\curvearrowright\\Gamma(H,W)$, which is the step that lets the largest structure of $H$ dominate $[Z]$.","marker":"[3]"},{"why":"It shows that every acylindrically hyperbolic group has a hyperbolic Cayley graph with an acylindrical non-elementary action, which is what makes $\\mathcal{AH}(G)$ non-trivial.","marker":"[10]"},{"why":"It provides the fact that every RAAG embeds as a finite-index normal subgroup of a RACG, used in the corollary.","marker":"[4]"},{"why":"It is the source of the comparison theorem that every RACG is $\\mathcal{AH}$-accessible, which the corollary relates to the extension-graph action.","marker":"[2]"}],"fun_headline_variants":["Finite extensions keep hyperbolic accessibility","Accessibility persists across finite extensions","Finite-index subgroups pass on accessibility","Hyperbolic structure survives finite extensions","Accessibility is finite-extension invariant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole comparison rests on the transfer lemma that turns the action of a finite-index subgroup on any hyperbolic Cayley graph of the overgroup into an equivalent action of the subgroup on its own hyperbolic Cayley graph; if that lemma failed for some structure, the proposed largest structure could not be compared with it.","fun_headline_variants_meta":{"raw":{"variants":["Finite extensions keep hyperbolic accessibility","Accessibility persists across finite extensions","Finite-index subgroups pass on accessibility","Hyperbolic structure survives finite extensions","Accessibility is finite-extension invariant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000471,"raw_usage":{"total_tokens":2271,"prompt_tokens":798,"completion_tokens":1473,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":414,"completion_tokens_details":{"reasoning_tokens":1415}},"tokens_in":414,"tokens_out":1473,"duration_ms":11641,"temperature":1.0,"reasoning_tokens":1415,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:30:17.267234+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a finitely generated group $G$, a normal finite-index subgroup $H$, and a hyperbolic or acylindrically hyperbolic generating set $Z$ of $G$ such that, with $[X]$ the largest structure of $H$, $\\sup_{x\\in X\\cup Y}|x|_Z=\\infty$. Since the proof would then fail at the domination step, such an example would refute the proposition.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference supplies the definitions of $\\mathcal{H}(G)$ and $\\mathcal{AH}(G)$, the transfer lemma used in both proofs, and the open problem that the paper answers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It proves the earlier result that $G$ is acylindrically hyperbolic under the same hypotheses, and its construction of $[X\\cup Y]$ is the starting point for the two propositions."},{"cited_title":"Extending group actions on metric spaces","cited_arxiv_id":"1703.03010","evidence_quote":"It is cited for the equivalence between the restricted action $H\\curvearrowright\\Gamma(G,Z)$ and an action $H\\curvearrowright\\Gamma(H,W)$, which is the step that lets the largest structure of $H$ dominate $[Z]$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It shows that every acylindrically hyperbolic group has a hyperbolic Cayley graph with an acylindrical non-elementary action, which is what makes $\\mathcal{AH}(G)$ non-trivial."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the fact that every RAAG embeds as a finite-index normal subgroup of a RACG, used in the corollary."},{"cited_title":"Largest acylindrical actions and stability in hierarchically hyperbolic groups","cited_arxiv_id":"1705.06219","evidence_quote":"It is the source of the comparison theorem that every RACG is $\\mathcal{AH}$-accessible, which the corollary relates to the extension-graph action."}],"review_version":1}