Finite extensions of $\mathcal{H}-$ and $\mathcal{AH}-$accessible groups

Sahana Balasubramanya

Finite extensions of $\mathcal{H}-$ and $\mathcal{AH}-$accessible groups

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.04894 v2 pith:N4P24FSV submitted 2019-08-14 math.GR

Sahana Balasubramanya This is my paper · ORCID

classification math.GR MSC 20F6520F6720E22

keywords acylindricallyhyperbolicstructuresAH-accessibilityH-accessibilityfiniteextensionsfinite-indexsubgroupslargestactiongeometricgrouptheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

Prop 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.

The 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.

This 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.

Recommendation: 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.

Referee Report

0 major / 6 minor

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].

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.

minor comments (6)

[§3, proof of Proposition 1.4, Eq. (1)] 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.
[§3, proof of Proposition 1.5, Eq. (4)] There is a spurious inverse in the displayed inequality: it should be |y^{-1}xy|_X, not |y^{-1}x^{-1}y|_X.
[§3, proof of Proposition 1.5, decomposition] 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.
[§3, proof of Proposition 1.5, last paragraph] 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.
[§3, proof of Proposition 1.4, coboundedness] 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).
[§3, proof of Proposition 1.4, use of [8, Lemma 6]] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the accessibility-transfer proofs do not assume their conclusions and rely only on independent lemmas.

full rationale

The central argument reduces the accessibility of G to the accessibility of a finite-index normal subgroup H. Given the largest structure [X] in AH(H) or H(H), the author constructs [X∪Y] and proves it is largest in AH(G) or H(G) by taking an arbitrary [Z] in the target poset, restricting the action to H, applying the Svarc-Milnor transfer lemma [1, Lemma 3.11] to obtain [W] in the corresponding poset for H, and using maximality of [X] to compare [W] with [X]. This is a direct proof, not a definitional equivalence: no equation identifies the conclusion with an input, and no fitted parameter is renamed as a prediction. The cited [1, Lemma 3.11] is a standard cobounded-action transfer result whose hypotheses (finite index, coboundedness, hyperbolicity/acylindricity) do not include accessibility, so citing it is independent support rather than load-bearing circularity. The fact that [1] is coauthored by the present author is irrelevant under the stated rules because the lemma is a genuine prior result, not an unverified premise that assumes the target theorem. Likewise [8, Lemma 6] supplies the non-trivial fact that [X∪Y]∈AH(G), but the paper's new content is the largest-element comparison, which is proved in detail for the acylindrical case. The hyperbolic case in Proposition 1.5 explicitly says the largest-element claim follows by the same argument with acylindricity dropped; this is a routine omitted proof, not a circular dependence. Corollary 3.1 inherits AH-accessibility from RAAGs via [1, Theorem 2.19(c)] and [4], both external to the present claim. No circular step was found.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests entirely on standard geometric group theory and on two external results: the Svarc-Milnor transfer lemma from [1] and the membership result [8, Lemma 6]. No free parameters, fitted constants, or ad hoc assumptions are introduced. The main proof is a direct calculation.

assumptions (6)

standard math Standard definitions of hyperbolic spaces, acylindrical actions, and Cayley graphs.
Used throughout Section 2; no independent evidence beyond standard background.
domain assumption The Svarc-Milnor transfer lemma [1, Lemma 3.11] that a cobounded action on a hyperbolic space yields an equivalent hyperbolic Cayley graph structure.
Invoked in Proposition 1.4 to obtain [W] ∈ AH(H) from the cobounded acylindrical action H ↷ Γ(G,Z). The lemma is cited, not reproved.
domain assumption The result [8, Lemma 6] that [X∪Y] is an acylindrically hyperbolic structure on G when [X] is the largest element of AH(H).
Used at the start of the proof of Proposition 1.4 to justify that the proposed largest element [X∪Y] belongs to AH(G).
domain assumption The action of Aut(H) on H(H) is well-defined and order preserving, and the largest element [X] of H(H) is fixed by all automorphisms.
Used in Proposition 1.5 to obtain the uniform conjugation bound (4). The order-preserving action is from [1, Section 2.3]; the invariance of the largest element follows from uniqueness.
domain assumption RAAGs are AH-accessible ([1, Theorem 2.19(c)]).
Used in Corollary 3.1 to conclude that a RACG containing a RAAG as a finite-index normal subgroup is AH-accessible.
domain assumption Every RAAG embeds as a finite-index normal subgroup of some RACG ([4, Lemma 3]).
Used in the remark after Corollary 3.1 to exhibit RACGs to which the corollary applies.

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Cite this review

Pith. "Pith review of Finite extensions of $\mathcal{H}-$ and $\mathcal{AH}-$accessible groups." pith.science (2026). https://pith.science/paper/N4P24FSV

@misc{pith2026190804894,
  author       = {Pith},
  title        = {Pith review of: Finite extensions of $\mathcalH-$ and $\mathcalAH-$accessible groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N4P24FSV}},
  note         = {Machine review of arXiv:1908.04894}
}

abstract

We prove that the group property of being $\mathcal{H}-$ and $\mathcal{AH}-$accessible is preserved under finite extensions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

C.Abbott, S.Balasubramanya, D.Osin; Hyperbolic struc tures on groups; Algebr. Geom. Topol. 19 (2019), no. 4, 17471835

work page 2019
[8]

A.Minasyan, D.Osin; Correction to: Acylindrical hyper bolicity of groups acting on trees; Math. Ann. 373 (2019), no. 1-2, 895900

work page 2019
[2]

C.Abbott, J.Behrstock, M.G.Durham; Largest acylindri cal actions and stability in Hi- erarchically hyperbolic groups; arXiv:1705.06219

work page Pith review arXiv
[3]

C.Abbott, D.Hume, D.Osin; Extending group actions on me tric spaces; arXiv:1703.03010, To appear in J. Topol. Anal

work page Pith review arXiv
[4]

M.W.Davis, T.Januszkiewicz; Right-angled Artin group s are commensurable with right-angled Coxeter groups; Journal of Pure and Applied Al gebra, Volume 153, Is- sue 3, (2000), 229 - 235

work page 2000
[5]

C.Drutu; Relatively hyperbolic groups: geometry and qu asi-isometric invariance; Com- ment. Math. Helv. 84 (2009), no. 3, 503546

work page 2009
[6]

M.Gersten, ed.); MSRI Publ

M.Gromov; Hyperbolic groups, in Essays in Group Theory ( G. M.Gersten, ed.); MSRI Publ. 8, 1987, pp. 75-263

work page 1987
[7]

A.Minasyan, D.Osin; Acylindrical hyperbolicity of gro ups acting on trees; Math. Ann. 362 (2015), no. 3-4, 10551105

work page 2015

Show all 10 references

[9]

D.Osin, Groups acting acylindrically on hyperbolic spa ces; arXiv:1712.00814v4

arXiv
[10]

D.Osin; Acylindrically hyperbolic groups; Trans. Ame r. Math. Soc. 368 (2016), no. 2, 851888. 7

2016

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

C.Abbott, S.Balasubramanya, D.Osin; Hyperbolic struc tures on groups; Algebr. Geom. Topol. 19 (2019), no. 4, 17471835

work page 2019

[2] [8]

A.Minasyan, D.Osin; Correction to: Acylindrical hyper bolicity of groups acting on trees; Math. Ann. 373 (2019), no. 1-2, 895900

work page 2019

[3] [2]

C.Abbott, J.Behrstock, M.G.Durham; Largest acylindri cal actions and stability in Hi- erarchically hyperbolic groups; arXiv:1705.06219

work page Pith review arXiv

[4] [3]

C.Abbott, D.Hume, D.Osin; Extending group actions on me tric spaces; arXiv:1703.03010, To appear in J. Topol. Anal

work page Pith review arXiv

[5] [4]

M.W.Davis, T.Januszkiewicz; Right-angled Artin group s are commensurable with right-angled Coxeter groups; Journal of Pure and Applied Al gebra, Volume 153, Is- sue 3, (2000), 229 - 235

work page 2000

[6] [5]

C.Drutu; Relatively hyperbolic groups: geometry and qu asi-isometric invariance; Com- ment. Math. Helv. 84 (2009), no. 3, 503546

work page 2009

[7] [6]

M.Gersten, ed.); MSRI Publ

M.Gromov; Hyperbolic groups, in Essays in Group Theory ( G. M.Gersten, ed.); MSRI Publ. 8, 1987, pp. 75-263

work page 1987

[8] [7]

A.Minasyan, D.Osin; Acylindrical hyperbolicity of gro ups acting on trees; Math. Ann. 362 (2015), no. 3-4, 10551105

work page 2015