Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups

Eduardo Mart\'inez-Pedroza; Ged Corob Cook; Ilaria Castellano; Shivam Arora

REVIEW 2 major objections 5 minor 40 references

Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In hyperbolic TDLC-groups of rational discrete cohomological dimension at most 2, every compactly presented closed subgroup is hyperbolic.

desk verdict 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. read the letter →

arxiv 1908.07946 v7 pith:MIWCRLBE submitted 2019-08-21 math.GR

Shivam Arora , Ilaria Castellano , Ged Corob Cook , Eduardo Martínez-Pedroza This is my paper · ORCID

classification math.GR MSC 20F6722D0520J0520E0620F65

keywords hyperbolicgroupstotallydisconnectedlocallycompacthomologicalfinitenesscohomologicaldimension2compactlypresentedweaklinearisoperimetricinequalitysmallcancellationCAT(-1)buildings

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

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.

What carries the argument

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 $FP_\infty$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $cd_{\mathbb{Q}}G \le \operatorname{asdim} G$.

Reading between the lines

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

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.

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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

Colleague,

The 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.

The 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.

Is 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.

Minor 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.

Verdict: 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[Section 5, Lemma 5.1] 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 3.3, Proposition 3.7] 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.

minor comments (5)

[Section 1, paragraph 1] There is a typo: 'posibble' should be 'possible'.
[Section 5, proof of Lemma 5.1] 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.
[Proof of Theorem 1.1] 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 4.6, proof of Proposition 4.7] There is a typo: 'it follows tha that' should be 'it follows that'.
[Title page] The title contains spacing errors: 'TOT ALL Y' and 'COMP ACT' should be 'TOTALLY' and 'COMPACT'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is proved from independently established characterizations and the discrete analogue is cited only as published background.

full rationale

The derivation of Theorem 1.1 is not circular. The proof chain is explicit: G hyperbolic implies, by Theorem 1.4, that G is compactly presented and satisfies the weak linear isoperimetric inequality; Theorem 1.5 transfers this inequality to the compactly presented closed subgroup H; Theorem 1.4 is then applied again to conclude H is hyperbolic. Each theorem is established inside the paper rather than assumed as input. Theorem 1.4 is proved from the Manning–Groves characterization [25] and the Rips-complex carry-over in Proposition 3.7; it does not assume the subgroup theorem. Theorem 1.5 is proved from the definition of the weak n-dimensional inequality, resolution-independence (Proposition 4.6), Lemma 5.1, and Proposition 4.5; none of these inputs states the conclusion of Theorem 1.5. The self-citations to [3], [16] and [17] supply foundations (rational discrete cohomology, finiteness properties, characterization of projectives) and the discrete analogue of the main theorem; they are published parameter-free results whose stated assumptions do not include the TDLC subgroup theorem, so under the review rules they count as independent evidence rather than circularity. The one serious concern raised against the paper is a proof-gap objection to Lemma 5.1: defining δ_n via the projection π∂_n may not guarantee ∂_n(Ω_n) ⊆ ker δ_{n-1}, so the displayed diagram might fail to commute. That is a correctness risk, not a circularity, because it does not exhibit a conclusion that is equivalent by construction to an input. Remark 5.2 itself flags that the mapping-cylinder argument works only for open subgroups, which is why Lemma 5.1 is needed; whether its proof is complete does not change the circularity verdict. Therefore no circular step is established.

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

No fitted constants or invented entities. The new concept 'weak n-dimensional linear isoperimetric inequality' is a definition within the existing framework, not a postulated physical or mathematical object with independent evidence requirements.

assumptions (6)

domain assumption The category Q[G]^dis is abelian with enough projectives, and a module is projective iff it is a direct summand of a proper permutation module (Proposition 2.2, cited to [17, Corollary 3.3]).
Underpins the definitions of type FP_n and cd_Q used throughout.
domain assumption A TDLC-group is compactly presented iff it admits a simply connected cellular G-complex with compact open cell stabilizers, finitely many G-orbits of cells in dimensions ≤ 2, and no cell inversions (Proposition 3.6, cited to [16, Proposition 3.4] and [10, I.8, Theorem 8.10]).
Connects compact presentability to the topological models used in Section 6.
domain assumption A hyperbolic TDLC-group acts on a finite-dimensional, contractible, locally finite simplicial complex with compact open stabilizers and finitely many orbits of cells (Proposition 3.7; proof carried over from the discrete case via Rips complexes on Cayley-Abels graphs).
Gives compact presentability and type FP∞ for hyperbolic TDLC-groups.
domain assumption Groves-Manning theorem: a simply connected 2-complex with uniformly bounded attaching maps satisfies the linear homological isoperimetric inequality iff its 1-skeleton is hyperbolic (Proposition 6.2, cited to [25]).
The bridge from the isoperimetric inequality to hyperbolicity in Theorem 1.4.
domain assumption Small cancellation theory: for amalgamated free products of profinite groups, a C'(1/12) symmetrized set R yields a C'(1/6) contractible 2-complex with hyperbolic 1-skeleton (Section 7, using [31], [36], [24]).
Establishes Theorem 7.1, the source of examples.
standard math Standard homological algebra: if a module has projective dimension ≤ d in an abelian category with enough projectives, then every d-th syzygy in a projective resolution is projective.
Used in the proof of Theorem 1.5 to conclude ker(∂_n) is projective.

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Pith. "Pith review of Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups." pith.science (2026). https://pith.science/paper/MIWCRLBE

@misc{pith2026190807946,
  author       = {Pith},
  title        = {Pith review of: Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MIWCRLBE}},
  note         = {Machine review of arXiv:1908.07946}
}

abstract

This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of the article: for hyperbolic TDLC-groups with rational discrete cohomological dimension $\leq 2$, hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group $\mathrm{Aut}(X)$ of a negatively curved locally finite $2$-dimensional building $X$ is a hyperbolic TDLC-group, whenever $\mathrm{Aut}(X)$ acts with finitely many orbits on $X$. Examples where this result applies include hyperbolic Bourdon's buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension $2$ when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

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Reference graph

Works this paper leans on

40 extracted references · 39 canonical work pages

[1]

Kompakt deﬁnierbare topologische Grupp en

Herbert Abels. Kompakt deﬁnierbare topologische Grupp en. Math. Ann. , 197:221–233, 1972

work page 1972
[2]

Homological and homotopical Dehn functions are diﬀerent

Aaron Abrams, Noel Brady, Pallavi Dani, and Robert Young . Homological and homotopical Dehn functions are diﬀerent. Proc. Natl. Acad. Sci. USA , 110(48):19206–19212, 2013

work page 2013
[3]

Subgroups of word hyperbolic groups in ra- tional dimension 2

Shivam Arora and Eduardo Mart ´ ınez-Pedroza. Subgroups of word hyperbolic groups in ra- tional dimension 2. Groups Geom. Dyn. , 2020

work page 2020
[4]

Totally disconnected, locally compac t groups as geometric objects

Udo Baumgartner. Totally disconnected, locally compac t groups as geometric objects. In Geometric group theory , pages 1–20. Springer, 2007

work page 2007
[5]

Hyperbolic groups have ﬂat- rank at most 1

Udo Baumgartner, R¨ ognvaldur G M¨ oller, and George A Willis. Hyperbolic groups have ﬂat- rank at most 1. Israel Journal of Mathematics , 190(1):365–388, 2012

work page 2012
[6]

Geometric characterization of ﬂat groups of automorphisms

Udo Baumgartner, G¨ unter Schlichting, and George A Will is. Geometric characterization of ﬂat groups of automorphisms. Groups Geom. Dyn. , 4(1):1–13, 2010

work page 2010
[7]

The boundary of negati vely curved groups

Mladen Bestvina and Geoﬀrey Mess. The boundary of negati vely curved groups. J. Amer. Math. Soc. , 4(3):469–481, 1991

work page 1991
[8]

Immeubles hyperboliques, dimension conf orme et rigidit´ e de Mostow

Marc Bourdon. Immeubles hyperboliques, dimension conf orme et rigidit´ e de Mostow. Geom. Funct. Anal., 7(2):245–268, 1997

work page 1997

Show all 40 references

[9]

Branched coverings of cubical complexes and subgroups of hyperbolic groups

Noel Brady. Branched coverings of cubical complexes and subgroups of hyperbolic groups. J. London Math. Soc. (2) , 60(2):461–480, 1999

1999
[10]

Bridson and Andr´ e Haeﬂiger

Martin R. Bridson and Andr´ e Haeﬂiger. Metric spaces of non-positive curvature , volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamen tal Principles of Mathemat- ical Sciences]. Springer-Verlag, Berlin, 1999

1999
[11]

Finitely presented simpl e groups and products of trees

Marc Burger and Shahar Mozes. Finitely presented simpl e groups and products of trees. C. R. Acad. Sci. Paris S´ er. I Math. , 324(7):747–752, 1997. 21

1997
[12]

Buyalo and Nina D

Serei V. Buyalo and Nina D. Lebedeva. Dimensions of loca lly and asymptotically self-similar spaces. Algebra i Analiz , 19(1):60–92, 2007

2007
[13]

Peter J. Cameron. Metric and topological aspects of the symmetric group of countable degree. European Journal of Combinatorics , 17(2-3):135–142, 1996

1996
[14]

Amenable hyperbolic groups

Pierre-Emmanuel Caprace, Yves de Cornulier, Nicolas M onod, and Romain Tessera. Amenable hyperbolic groups. J. Eur. Math. Soc. (JEMS) , 017(11):2903–2947, 2015

2015
[15]

Rational discrete ﬁrst degree coho mology for totally disconnected locally compact groups

Ilaria Castellano. Rational discrete ﬁrst degree coho mology for totally disconnected locally compact groups. Math. Proc. Cambridge Philos. Soc. , 168(2):361–377, 2020

2020
[16]

Finiteness propert ies of totally disconnected locally compact groups

Ilaria Castellano and G Corob Cook. Finiteness propert ies of totally disconnected locally compact groups. Journal of Algebra , 543:54–97, 2020

2020
[17]

Ilaria Castellano and Thomas S. W eigel. Rational discr ete cohomology for totally disconnected locally compact groups. J. Algebra , 453:101–159, 2016

2016
[18]

Metric geometry of locally compact groups , volume 25 of EMS Tracts in Mathematics

Yves Cornulier and Pierre de la Harpe. Metric geometry of locally compact groups , volume 25 of EMS Tracts in Mathematics . European Mathematical Society (EMS), Z¨ urich, 2016. Winner of the 2016 EMS Monograph Award

2016
[19]

Fleming and Eduardo Mart ´ ınez-Pedroza

Joshua W. Fleming and Eduardo Mart ´ ınez-Pedroza. Fini teness of homological ﬁlling func- tions. Involve, 11(4):569–583, 2018

2018
[20]

Steve M. Gersten. Reducible diagrams and equations ove r groups. In Essays in group theory , volume 8 of Math. Sci. Res. Inst. Publ. , pages 15–73. Springer, New York, 1987

1987
[21]

Steve M. Gersten. A cohomological characterization of hyperbolic groups. 1996

1996
[22]

Steve M. Gersten. Subgroups of word hyperbolic groups i n dimension 2. J. London Math. Soc. (2) , 54(2):261–283, 1996

1996
[23]

Steve M. Gersten. Cohomological lower bounds for isope rimetric functions on groups. Topol- ogy, 37(5):1031–1072, 1998

1998
[24]

Gersten and H

Steve M. Gersten and H. B. Short. Small cancellation the ory and automatic groups. Invent. Math., 102(2):305–334, 1990

1990
[25]

Daniel Groves and Jason F. Manning. Dehn ﬁlling in relat ively hyperbolic groups. Israel J. Math., 168:317–429, 2008

2008
[26]

Hanlon and Eduardo Mart ´ ınez-Pedroza

Richard G. Hanlon and Eduardo Mart ´ ınez-Pedroza. Lifting group actions, equivariant towers and subgroups of non-positively curved groups. Algebr. Geom. Topol., 14(5):2783–2808, 2014

2014
[27]

Hanlon and Eduardo Mart ´ ınez Pedroza

Richard G. Hanlon and Eduardo Mart ´ ınez Pedroza. A subg roup theorem for homological ﬁlling functions. Groups Geom. Dyn. , 10(3):867–883, 2016

2016
[28]

Realisation and dismantlability

Sebastian Hensel, Damian Osajda, and Piotr Przytycki. Realisation and dismantlability. Geom. Topol., 18(4):2079–2126, 2014

2014
[29]

Boundaries of hyperbo lic groups

Ilya Kapovich and Nadia Benakli. Boundaries of hyperbo lic groups. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) , volume 296 of Contemp. Math., pages 39–93. Amer. Math. Soc., Providence, RI, 2002

2000
[30]

M¨ oller

Bernhard Kr¨ on and R¨ ognvaldur G. M¨ oller. Analogues of Cayley graphs for topological groups. Math. Z. , 258(3):637–675, 2008

2008
[31]

Lyndon and Paul E

Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory . Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition

2001
[32]

Subgroups of relatively hyperbolic groups of Bredon cohomolog- ical dimension 2

Eduardo Mart ´ ınez-Pedroza. Subgroups of relatively hyperbolic groups of Bredon cohomolog- ical dimension 2. J. Group Theory , 20(6):1031–1060, 2017

2017
[33]

A model for the univer sal space for proper actions of a hyperbolic group

David Meintrup and Thomas Schick. A model for the univer sal space for proper actions of a hyperbolic group. New York J. Math. , 8:1–7 (electronic), 2002

2002
[34]

Bounded cohomology characterizes hyper bolic groups

Igor Mineyev. Bounded cohomology characterizes hyper bolic groups. Q. J. Math. , 53(1):59– 73, 2002

2002
[35]

Structure theory of totally dis connected locally compact groups via graphs and permutations

R¨ ognvaldur G M¨ oller. Structure theory of totally dis connected locally compact groups via graphs and permutations. Canadian Journal of Mathematics , 54(4):795–827, 2002

2002
[36]

A. Yu. Ol ′ shanski ˘ ı.Geometry of deﬁning relations in groups , volume 70 of Mathematics and its Applications (Soviet Series) . Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1989 Russian original by Yu. A. Bakhturi n

1991
[37]

Galois cohomology, corrected repr int of the 1997 english edition

Jean-Pierre Serre. Galois cohomology, corrected repr int of the 1997 english edition. Springer Monographs in Mathematics, Springer-Verlag, Berlin , pages 94720–3840, 2002

1997
[38]

Trivalent polygonal complexes of nonpositive cu rvature and Platonic sym- metry

Jacek ´Swi ‘atkowski. Trivalent polygonal complexes of nonpositive cu rvature and Platonic sym- metry. Geom. Dedicata, 70(1):87–110, 1998

1998
[39]

Zur topologischen Algebra

David Van Dantzig. Zur topologischen Algebra. III. Bro uwersche und Cantorsche Gruppen. Compositio Math. , 3:408–426, 1936

1936
[40]

The structure of totally disconnected, locally compact groups

George Willis. The structure of totally disconnected, locally compact groups. Math. Ann. , 300(2):341–363, 1994. 22 S. ARORA, I. CASTELLANO, G. COROB COOK, AND E. MART ´INEZ-PEDROZA Email address : sarora17@mun.ca Email address : ilaria.castellano88@gmail.com Email address : g...

1994

Pith tools

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

[1] [1]

Kompakt deﬁnierbare topologische Grupp en

Herbert Abels. Kompakt deﬁnierbare topologische Grupp en. Math. Ann. , 197:221–233, 1972

work page 1972

[2] [2]

Homological and homotopical Dehn functions are diﬀerent

Aaron Abrams, Noel Brady, Pallavi Dani, and Robert Young . Homological and homotopical Dehn functions are diﬀerent. Proc. Natl. Acad. Sci. USA , 110(48):19206–19212, 2013

work page 2013

[3] [3]

Subgroups of word hyperbolic groups in ra- tional dimension 2

Shivam Arora and Eduardo Mart ´ ınez-Pedroza. Subgroups of word hyperbolic groups in ra- tional dimension 2. Groups Geom. Dyn. , 2020

work page 2020

[4] [4]

Totally disconnected, locally compac t groups as geometric objects

Udo Baumgartner. Totally disconnected, locally compac t groups as geometric objects. In Geometric group theory , pages 1–20. Springer, 2007

work page 2007

[5] [5]

Hyperbolic groups have ﬂat- rank at most 1

Udo Baumgartner, R¨ ognvaldur G M¨ oller, and George A Willis. Hyperbolic groups have ﬂat- rank at most 1. Israel Journal of Mathematics , 190(1):365–388, 2012

work page 2012

[6] [6]

Geometric characterization of ﬂat groups of automorphisms

Udo Baumgartner, G¨ unter Schlichting, and George A Will is. Geometric characterization of ﬂat groups of automorphisms. Groups Geom. Dyn. , 4(1):1–13, 2010

work page 2010

[7] [7]

The boundary of negati vely curved groups

Mladen Bestvina and Geoﬀrey Mess. The boundary of negati vely curved groups. J. Amer. Math. Soc. , 4(3):469–481, 1991

work page 1991

[8] [8]

Immeubles hyperboliques, dimension conf orme et rigidit´ e de Mostow

Marc Bourdon. Immeubles hyperboliques, dimension conf orme et rigidit´ e de Mostow. Geom. Funct. Anal., 7(2):245–268, 1997

work page 1997