Cohomology of group theoretic Dehn fillings II

Bin Sun; Nansen Petrosyan

Cohomology of group theoretic Dehn fillings II

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that for sufficiently deep Dehn fillings of a hyperbolically embedded subgroup, the relative cohomology of the group pair is isomorphic to that of the filled pair, and a spectral sequence computes the absolute cohomology…

desk verdict Strong, careful paper with a genuinely new spectral sequence for Dehn filling cohomology; the main load-bearing input is an external published theorem, and that dependency is handled cleanly. read the letter →

arxiv 1908.01290 v6 pith:LFV27V2F submitted 2019-08-04 math.GR

Nansen Petrosyan , Bin Sun This is my paper · ORCID

classification math.GR MSC 20F6720F1020E06

keywords groupcohomologyDehnfillinghyperbolicallyembeddedsubgroupCohen-LyndonpropertyrelativePoincarédualitypairsimplicialvolumeacylindricallyhyperbolic

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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 for a group $G$ with a hyperbolically embedded subgroup $H$, when $N$ is a sufficiently deep normal subgroup of $H$, the cohomology of the Dehn filling quotient $G/\langle\langle N\rangle\rangle$ is controlled by a spectral sequence built from the cohomology of $H$ and $N$. It also establishes an algebraic form of excision: the relative cohomology of $(G,H)$ is isomorphic to that of the filled pair $(\overline{G}, \overline{H})$. This yields explicit bounds on cohomological dimension, preservation of finiteness properties $FP_n$, and applications to Poincaré duality groups and simplicial volume.

What carries the argument

The key object is the Cohen–Lyndon triple: a group triple $(G,\{H_\lambda\},\{N_\lambda\})$ for which the normal closure of the $N_\lambda$'s is a free product of conjugates $t N_\lambda t^{-1}$ over a transversal. This structural decomposition, quoted from the companion paper, permits an identification of $H^q(\langle\langle N\rangle\rangle;A)$ with a co-induced product, making the spectral sequence of Theorem 4.2 computable. That spectral sequence, together with its morphism to the Lyndon–Hochschild–Serre spectral sequence, is the engine that yields the cohomological decomposition and the excision isomorphism.

What would settle it

One could look for a hyperbolically embedded subgroup $H$ in a group $G$ and a sufficiently deep normal subgroup $N$ of $H$ such that the normal closure of $N$ in $G$ is not a free product of conjugates of $N$; if such an example exists, the main spectral sequence and its consequences would collapse.

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Extended reading notes

Core claim

The central discovery is that the Cohen–Lyndon structure of the normal closure $\langle\langle N\rangle\rangle$ allows one to identify the cohomology of $\langle\langle N\rangle\rangle$ with a direct sum of co-induced modules, leading to a morphism of Lyndon–Hochschild–Serre spectral sequences that becomes an isomorphism on the $E^2$ page for $q>0$. Consequently, for all sufficiently deep $N$, the relative cohomology $H^n(G,H;A)$ is naturally isomorphic to $H^n(\overline{G},\overline{H};A)$ for every $G$-module $A$. This algebraic excision underlies the paper's applications to Poincaré duality pairs and to quotients of acylindrically hyperbolic groups.

Load-bearing premise

The entire argument rests on the Cohen–Lyndon property for sufficiently deep Dehn fillings, a theorem imported from a companion paper; if that structural fact failed, the spectral sequence and all later results would not follow.

Editorial extensions

If this is right

If the main theorem holds, then for sufficiently deep Dehn fillings the cohomological dimension of the quotient is bounded by the maximum of the original group's dimension, the peripheral dimension plus one, and the filled peripheral dimension.
The algebraic excision implies that any property detected by relative cohomology is invariant under sufficiently deep Dehn fillings, so in particular whether a pair is a Poincaré duality pair is preserved under appropriate fillings.
Theorem B shows that when the original pair is a Poincaré duality pair and the filled peripheral subgroups are Poincaré duality groups, the filled quotient is a Poincaré duality group.
Theorem C gives that the simplicial volume of the filled quotient is bounded above by the relative simplicial volume, and positivity follows when the group is hyperbolic relative to the filled peripheral subgroups.
Theorems D and E provide acylindrically hyperbolic quotients with prescribed cohomology and finiteness properties, including quotients with Property (T) and specific $FP_n$ thresholds.

Reading between the lines

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

If the algebraic excision holds, it suggests that Dehn fillings behave like topological surgery in cohomology, so one might expect analogous computations for other cohomology theories such as bounded cohomology or $L^2$ cohomology.
The spectral sequence might be extendable to compute other invariants under Dehn filling, such as the growth of cohomology with twisted coefficients, which would further clarify the structure of Dehn-filled groups.
The reliance on the Cohen–Lyndon property points to a potential program: proving analogous decomposition theorems for wider classes of fillings could extend these cohomological results to graphs of groups or other relatively hyperbolic settings.

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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

I agree with the reader's verdict: this is a strong paper that deserves a serious referee. The genuinely new content is the spectral sequence in Theorem A(i) and the relative cohomology excision in Theorem A(ii), both built on the Cohen-Lyndon property from the companion paper [Sun20]. The authors are upfront about that dependency, and it is the right division of labor: [Sun20] is published, the statement they quote is explicit, and they correctly convert 'sufficiently deep' to the finite-set formulation (Remark 3.18) so the hypotheses line up. The coefficient-module caveat in Theorem A, that N acts trivially on A, is stated and is exactly what the spectral sequence needs for the q=1 row.

What the paper does well beyond Theorem A is the range of applications. Theorem B preserves Poincare duality under Dehn fillings, and Theorem C gets simplicial volume bounds, generalizing Fujiwara-Manning. Theorems D and E add cohomological control to earlier SQ-universality and common quotient results, and Section 8 shows the machinery has teeth: quotients with property (T) and prescribed FP_n gaps, and a negative answer to an actual Betti number question. Example 4.9 gives a clean sharpness example for the dimension bound in Theorem A(iii). The proof structure is clear: Proposition 4.1 and the spectral sequence comparison in Theorem 4.2 do the heavy lifting, and the later sections are honest applications.

Soft spots: the biggest one is the dependence on [Sun20, Theorem 5.1]. If that theorem had a gap, this paper would inherit it. That is a dependency, not a defect, and the stress-test confirms the usage is correct. The paper is long and dense, especially Sections 7.2 and 7.3, but the technical work there is aimed at a real purpose. I did not find any place where the argument hides a step or overclaims. The citation pattern looks fair: the relevant prior work by DGO, Hull, Wang, and Fujiwara-Manning is credited, and the companion paper is clearly flagged.

Who is this for? Geometric group theorists and people who work on cohomological finiteness properties. A reader comfortable with spectral sequences and relatively hyperbolic groups will get a lot out of it. I would send it to peer review without hesitation, and I would not be surprised if it lands in a good journal. If I were a referee I would check the [Sun20] dependency and the proof of Theorem 5.2, but I see no reason to expect a problem there.

Referee Report

0 major / 5 minor

Summary. The manuscript develops a cohomological framework for group-theoretic Dehn fillings. The main result, Theorem A, gives, for sufficiently deep normal subgroups N of a hyperbolically embedded subgroup H in G, a spectral sequence computing H^*(Gbar; A) from H^*(Hbar; H^*(N; A)) and H^*(G; A), an excision-type isomorphism between the relative cohomology of (G,H) and (Gbar,Hbar), a direct-sum decomposition of H^n(Gbar; A) for large n, a cohomological dimension bound, and an FP_n criterion. The paper then applies this framework to prove preservation of Poincaré duality under Dehn fillings (Theorem B), simplicial volume bounds (Theorem C), and theorems on common quotients of acylindrically hyperbolic groups with controlled cohomology (Theorems D and E), followed by several striking applications in Section 8. The proofs are built on the Cohen-Lyndon property for sufficiently deep Dehn fillings, quoted as Theorem 3.22 from Sun's companion paper [Sun20].

Significance. If the results hold, this is a substantial contribution to the cohomological study of Dehn fillings. Theorem A provides a genuine spectral sequence tool for Dehn fillings, and the excision isomorphism is an algebraic analog of topological excision that is used to obtain new results on Poincaré duality pairs, simplicial volume, and cohomological finiteness of acylindrically hyperbolic quotients. The paper is careful and detailed: the spectral sequence arguments in Section 4 are explicit, and Example 4.9 shows that the dimension threshold in Theorem A(iii) is sharp. The main external input, the Cohen-Lyndon property, is cited from a published companion paper and is applied with the correct deepness and coefficient hypotheses; I see no circularity or hidden parameter-fitting. The presentation is generally clear, and the applications are well motivated.

minor comments (5)

[Section 4, Proposition 4.1] The parenthetical in the statement of Proposition 4.1 is ambiguous: it reads "for any G-module A and q > 1 (also for any G-module A and q > 0)", but the proof establishes q = 1 only under the additional hypothesis that <N> = <langle>N<rangle> acts trivially on A. This should be reformulated, for example as "for q > 1, and also for q > 0 when A is trivial as a <N>-module".
[Section 7.2, Lemma 7.6] The displayed inequality in the proof of Lemma 7.6, "2n * 50D <= 3nD", does not match the counting: there are 2l components, each of p_ell-length at least 50D, bounding a 3l-gon. The printed inequality appears to contain a typographical error and should be something like "2l * 50D <= 3lD"; the contradiction is unaffected.
[Section 7.1, proof of Theorem D] In the proof of statement (v), the phrase "in which case (iii) is a void statement" should refer to statement (v): when the original group G has torsion, cd(G) = infinity, so the desired inequality cd(Gbar) <= max{cd(G), cd(C)} is vacuous. Statement (iii) concerns torsion-freeness and is not the statement being justified at that point.
[Section 5, Theorem 5.2] In the proof of Theorem 5.2, the vanishing H^1(G; ZG) = 0 is used without explanation. It follows from the fact that (G,H) is a PD(n)-pair with n >= 3, which forces G to be a duality group of dimension n-1; adding a sentence with this justification would improve clarity.
[Section 8.3, Corollary 8.5] In the remark after Corollary 8.5, "the quotients {G_k}_{k>=1}" should be "{G_k}_{k>=2}", consistent with the statement of the corollary.

Circularity Check

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No significant circularity: the central Cohen-Lyndon input is an external published theorem, and the spectral sequence/excision results are derived from it by standard algebra.

full rationale

The derivation chain is not circular. The paper's central input, Theorem 3.22 (Sun [Sun20, Thm 5.1]), is quoted verbatim as an external structural result: for sufficiently deep normal subgroups, the triple (G, {H_lambda}, {N_lambda}) is Cohen-Lyndon. The paper does not redefine its target in terms of this theorem: the spectral sequence in Theorem A(i) and the excision isomorphism in Theorem A(ii) are obtained by standard homological algebra (Bass-Serre tree, Shapiro's lemma, Lyndon-Hochschild-Serre spectral sequence, kernel spectral sequence) from the free-product decomposition x<N>y = * t N_lambda t^{-1}. No parameter is fitted to the data being 'predicted'; the constants in 'sufficiently deep' are existential and are converted from relative-metric to finite-set form by local finiteness (Remark 3.18). The only self-citation is [Sun20], and while it is load-bearing as a premise, it is a published, parameter-free theorem whose assumptions do not include the target results; under the stated review rules this counts as independent support rather than circularity. The paper also flags its own limitations (e.g., Theorem A(iii) fails below dimension cd(H)+2, Example 4.9), which further indicates the claims are not being forced by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the external Cohen-Lyndon property [Sun20] and standard cohomological machinery. No free parameters are fitted and no new entities are postulated. The only paper-specific assumption is the hyperbolically embedded subgroup setup inherited from DGO17.

assumptions (3)

domain assumption The Cohen-Lyndon property for sufficiently deep Dehn fillings of hyperbolically embedded subgroups (Sun20, Theorem 5.1).
This external theorem guarantees that the normal closure of N decomposes as a free product of conjugates tN_λ t^{-1}, which is the structural input for the spectral sequence and excision in Section 4.
domain assumption The definition and basic properties of (weakly) hyperbolically embedded subgroups from DGO17.
Used throughout, e.g., Theorem 3.20 preserves hyperbolic embedding under Dehn filling and Proposition 3.9 transitivity of hyperbolic embedding.
standard math Standard tools in group cohomology: Lyndon-Hochschild-Serre spectral sequence, Shapiro's lemma, Bieri-Eckmann duality, Stallings-Swan theorem, and finiteness criteria of Bieri and Brown.
These are unproved background results invoked in Sections 4, 5, and 7, including Theorem 3.1, Corollary 4.6, and the duality pair machinery.

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Pith. "Pith review of Cohomology of group theoretic Dehn fillings II." pith.science (2026). https://pith.science/paper/LFV27V2F

@misc{pith2026190801290,
  author       = {Pith},
  title        = {Pith review of: Cohomology of group theoretic Dehn fillings II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LFV27V2F}},
  note         = {Machine review of arXiv:1908.01290}
}

abstract

We study the cohomology of group theoretic Dehn fillings. Applying the Cohen-Lyndon property for sufficiently deep Dehn fillings of hyperbolically embedded subgroups $H\hookrightarrow_h G$, obtained by the second named author, we derive a spectral sequence that computes the cohomology of the corresponding Dehn filling quotients $\overline{G}$. As an application, we establish an isomorphism between the relative cohomology of the group pair $(G, H)$ and its sufficiently deep Dehn filling quotient pair $(\overline{G}, \overline{H})$. This allows us to generalise the results of Fujiwara and Manning on simplicial volume of Dehn fillings of hyperbolic manifolds to Dehn fillings of Poincar\'e duality pairs. We also strengthen the results of Olshanskii, Dahmani-Guirardel-Osin and Hull on SQ-universality and common quotients of acylindrically hyperbolic groups by adding cohomological finiteness conditions. We apply these results to obtain hyperbolic and acylindrically hyperbolic quotients with special properties.

Figures

Figures reproduced from arXiv: 1908.01290 by the authors.

**Figure 1.** Illustration of H ãÑh G (a) H ãÑwh pG, Gq for every subgroup H ď G. (b) H ãÑh pG, Gq for every finite subgroup H ď G. (c) G ãÑh pG, Hq. (d) If G can be decomposed as a free product of its subgroups tGλuλPΛ (denoted by G “ ˚λPΛGλ), then tGλuλPΛ ãÑh pG, Hq [DGO17, Example 4.12]. (e) More generally, suppose that G “ π1pGq, where G is a graph of groups. Let tGvuvPV G be the collection of vertex subgroups and tGeuePEG th… view at source ↗

**Figure 2.** Mapping cylinder CpLi , Tiq Given a closed oriented n-manifold M possibly with boundary BM, the simplicial volume of pM, BMq is defined as ||M, BM|| “ ||rM, BMs||1 where rM, BMs P HnpM, BM; Rq is the image of the fundamental class under the change of coefficients map HnpM, BM;Zq Ñ HnpM, BM; Rq. The following result is again a generalisation of [FM11, Theorem 1.5]. Corollary 6.6. Let M be a compact oriented n-manifo… view at source ↗

**Figure 3.** Some possible configurations of the two pairs of adjacent components of p Note that α lies in the 2-neighborhood of the orbit Hλ, and γ lies in the R-neighborhood of α. Thus, γ lies in the pR ` 2q-neighborhood of Hλ. For λ, µ P Λ, the orbits Hλ and Hµ are subsets of ΓpG, X\Kq. Thus, it makes sense to talk about the diameter of HµXpgHλq `ǫ in ΓpX\Kq, which is denoted by diam ` Hµ X pgHλq `ǫ ˘ . Lemma 7.9. For every … view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Case 1 Case 1 is displayed by [PITH_FULL_IMAGE:figures/full_fig_p033_4.png]

**Figure 5.** Figure 5: Case 2 Case 2 is displayed by [PITH_FULL_IMAGE:figures/full_fig_p033_5.png]

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Pith tools

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

[1] [1]

Antol \' n, R

Y . Antol \' n, R . C oulon, and G . G andini. Farrell- J ones via D ehn fillings. J. Topol. Anal. , 10(4):873--895, 2018

2018

[2] [2]

Aramayona, D

J. Aramayona, D. Degrijse, C. Mart\' nez-P\' e rez, and J. Souto. Geometric dimension of lattices in classical simple L ie groups. J. Topol. , 10(2):632--667, 2017

work page 2017

[3] [3]

I. Agol, D. Groves, and J. Manning. An alternate proof of W ise's malnormal special quotient theorem. Forum Math. Pi , 4:e1, 54, 2016

2016

[4] [4]

I. Agol. The virtual H aken conjecture. Doc. Math. , 18:1045--1087, 2013. With an appendix by I. Agol, D. Groves, and J. Manning

2013

[5] [5]

J. Alonso. Finiteness conditions on groups and quasi-isometries. J. Pure Appl. Algebra , 95(2):121--129, 1994

work page 1994

[6] [6]

A. Ash. Small-dimensional classifying spaces for arithmetic subgroups of general linear groups. Duke Math. J. , 51(2):459--468, 1984

work page 1984

[7] [7]

Bieri and B

R. Bieri and B. Eckmann. Relative homology and P oincar\' e duality for group pairs. J. Pure Appl. Algebra , 13(3):277--319, 1978

1978

[8] [8]

Bestvina and M

M. Bestvina and M. Feighn. A hyperbolic Out(F_n) -complex. Groups Geom. Dyn. , 4(1):31--58, 2010

work page 2010