The McCullough-Miller complex for right angled Artin groups

Conchita Mart\'inez P\'erez; Peio Ardaiz Gal\'e

REVIEW 3 major objections 5 minor 25 references

The McCullough-Miller complex for right angled Artin groups

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper builds a contractible complex for pure symmetric automorphism groups of every right-angled Artin group, and uses it to compute cohomological dimension and ℓ²-Betti numbers.

desk verdict A valuable RAAG analogue of McCullough–Miller with a real gap in the Existence Lemma; Theorem B is unproved as written but the construction deserves a careful referee. read the letter →

arxiv 2506.03377 v2 pith:NJQ3LKYM submitted 2025-06-03 math.GR

Peio Ardaiz Galé , Conchita Martínez Pérez This is my paper

classification math.GR MSC 20J0620F3657M0755P20

keywords right-angledArtingroupspuresymmetricautomorphismsMM-typecomplexWhiteheadposetpartialconjugationscohomologicaldimensionℓ2-Bettinumbers

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

Right-angled Artin groups interpolate between free groups and free abelian groups, and their pure symmetric outer automorphism groups $\Sigma\mathrm{POut}(A_\Gamma)$ generalize the objects studied for free groups. The paper's central claim is that for every finite simplicial graph $\Gamma$ there is a simplicial complex $\mathrm{MM}_\Gamma$, built from the $\Gamma$-Whitehead poset, on which $\Sigma\mathrm{POut}(A_\Gamma)$ acts with free abelian cell stabilizers and with fundamental domain the simplicial realization of the Whitehead poset, and that $\mathrm{MM}_\Gamma$ is contractible. The value of that claim is that these groups gain a Deligne-type complex, the kind of object that has powered cohomological-dimension, cohomology-ring, and $\ell^2$-Betti-number computations for free groups. The paper also proves a formula for the cohomological dimension of $\Sigma\mathrm{POut}(A_\Gamma)$ in terms of the rank of Whitehead elements, and matches von Neumann dimensions of the complex's cohomology with those of the non-nuclear part of the Whitehead poset.

What carries the argument

The central object is the $\Gamma$-Whitehead poset $\mathrm{Wh}_\Gamma$: its elements are $\Gamma$-vertex types, i.e. families of pairwise compatible based partitions of the generator set whose petals are unions of connected components of $\Gamma$ minus the star of the operative generator. The complex $\mathrm{MM}_\Gamma$ is the simplicial realization of the poset of marked vertex types, equivalently the coset poset of the stabilizers of marked types. The proof that it is contractible proceeds by adding stars of nuclear vertices in increasing height, measured by cyclic word lengths of a fixed finite word set; the notion of reductivity records how much an automorphism decreases that height. The technical engine is an adjacency-counter formula for the length change of a partial conjugation, a peak-reduction statement for automorphisms of right-angled Artin groups, and an Existence Lemma producing a strictly reductive partial conjugation when height is not minimal. The final homotopy equivalences come from refinement and disjunction operations on based partitions and from standard poset lemmas for cone decompositions.

What would settle it

One concrete check is to compute, for the edgeless graph with four vertices and the word $w = x^2 b x^{-2} c$ used in the paper's Example 6.13, whether a peak-reduction factorization of a height-lowering $\beta^{-1}$ with respect to $W' = W \cup \{b^k\}$ can have a first factor $\alpha_1$ with $\mathrm{red}_{W'}(\alpha_1^{-1}) > 0$ but $\mathrm{red}_W(\alpha_1^{-1}) \leq 0$; if such a factor exists, the displayed equality in Lemma 6.16 fails and the contractibility proof needs a missing argument.

Watch

Extended reading notes

Core claim

The paper's discovery is that the classical free-group complex construction survives when the graph $\Gamma$ is arbitrary, provided the petals of the partitions are required to be unions of connected components of $\Gamma - \mathrm{st}(a)$, and compatibility is defined through a crossing condition. With this added structure, the marked poset is still a coset poset for stabilizers generated by partial conjugations, and the stabilizers are free abelian. Theorem A states that $\Sigma\mathrm{POut}(A_\Gamma)$ acts on $\mathrm{MM}_\Gamma$ with free abelian cell stabilizers and a strong fundamental domain. Theorem B, the main result, asserts that $\mathrm{MM}_\Gamma$ is contractible, making it a Deligne-type complex for $\Sigma\mathrm{POut}(A_\Gamma)$. The paper then proves that the cohomological dimension of $\Sigma\mathrm{POut}(A_\Gamma)$ equals the maximum rank $r(A)$ over the $\Gamma$-Whitehead poset, realized by a free abelian subgroup, and that the von Neumann dimensions of the cohomology of $\mathrm{MM}_\Gamma$ match those determined by the non-nuclear part of the Whitehead poset, with a stronger $\ell^2$-cohomology statement under an extra centralizer condition.

Load-bearing premise

The contractibility proof rests on the Existence Lemma, which assumes that after adjoining powers $b^k$ of every generator to the word set and peak-reducing a height-lowering automorphism, the first factor is a partial conjugation whose length decrease is already positive on the original word set; the displayed equality comparing the two length functions is not justified, because the extra $b^k$ powers can change by unequal amounts under partial conjugations.

Editorial extensions

If this is right

The group $\Sigma\mathrm{POut}(A_\Gamma)$ gets a contractible complex with free abelian stabilizers, so $\mathrm{MM}_\Gamma$ can serve as a Deligne-type model for computing its cohomology.
The cohomological dimension of $\Sigma\mathrm{POut}(A_\Gamma)$ is exactly the maximum rank $r(A)$ over the $\Gamma$-Whitehead poset, and this dimension is realized by a free abelian subgroup.
The von Neumann dimension of $H^i(\mathrm{MM}_\Gamma; \mathcal{N}(\Sigma\mathrm{POut}(A_\Gamma)))$ equals that of $\mathcal{N}(\Sigma\mathrm{POut}(A_\Gamma)) \otimes H^{i-1}(|\mathrm{Wh}_\Gamma^0|)$, and under the centralizer condition the $\ell^2$-cohomology decomposes as $\ell^2(G) \otimes H^{i-1}(|\mathrm{Wh}_\Gamma^0|)$.
For an edgeless graph, $\mathrm{MM}_\Gamma$ specializes to the classical free-group complex, so the results reproduce the known statements for $\Sigma\mathrm{POut}(F_n)$.
Since every right-angled Artin group is isomorphic to $\Sigma\mathrm{POut}(A_\Delta)$ for some graph $\Delta$, the construction gives a uniform Deligne-type model covering all right-angled Artin groups.

Reading between the lines

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

Because all cell stabilizers are free abelian, the action makes $\mathrm{MM}_\Gamma$ a natural model for the classifying space for abelian subgroups of $\Sigma\mathrm{POut}(A_\Gamma)$, so it can feed equivariant cohomology computations beyond ordinary cohomological dimension.
The height filtration used in the contractibility proof is a natural place to attach the homological finiteness invariants of the group, in analogy with how the free-group complex was used for those invariants; the paper does not pursue this.
The extra centralizer condition that separates the two parts of Theorem D is already noted in the paper to fail in general, so the weaker von Neumann-dimension statement is likely the right general formulation; identifying exactly which graphs satisfy the stronger condition would be a concrete follow-up.
A comparison with cube-complex models for the larger symmetric outer automorphism group would clarify whether the poset-based model and the cube-based model carry the same equivariant cohomology.

Share X Bluesky LinkedIn Reddit HN

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

Quick take: this is a natural and worthwhile generalization of McCullough-Miller to RAAGs. The Gamma-Whitehead poset, the marked equivalence relation, and the complex MM_Gamma are genuinely new, and the paper is honest about Corrigan's independent proof of Theorem C. But the proof of Theorem B leans on Lemma 6.16 (the Existence Lemma), and the proof of that lemma has a gap you need to know before citing it.

What is good. Theorem A is proven cleanly: Lemma 5.3 gives free abelian stabilizers, the compatibility analysis in Section 5 is careful, and the Fundamental Domain description is convincing. The factorization lemma and the adjacency-counter formalism are sensible, and Example 6.13 is a real service: it identifies a genuine problem in McCullough-Miller's original Lemma 3.8. The citation practice is honest: Theorem C is explicitly credited to Corrigan, and the relation to Abgrall's work is stated.

Where it slips. Lemma 6.16. The proof adds W' = W ∪ {b^k | b in VΓ} and peak-reduces beta^{-1} with respect to W'. It wants to show the first factor alpha_1 is a strictly reductive partial conjugation. Two things are not justified as written. First, the displayed equality red_W(alpha_1^{-1}) = ||u0||_W - ||alpha_1^{-1}(u0)||_W = ||u0||_{W'} - ||alpha_1(u0)||_{W'} does not follow: it shifts from alpha_1^{-1} to alpha_1, and it treats the added b^k words as if they canceled uniformly. For a partial conjugation C_a^A, each b in A contributes -2 to red_{W'}, so the correct relation is red_W(alpha_1^{-1}) = red_{W'}(alpha_1^{-1}) + 2|A|. That inequality would actually save the implication from red_{W'} > 0 to red_W > 0, but the proof never states or proves it. Second, the argument that alpha_1 must be a partial conjugation asks to choose k >= ||v||_{W'}, which is circular because W' depends on k; the peak reduction is done after W' is fixed, and you cannot then choose k to rule out the first factor.

Where that leaves us. Lemma 7.9 and Proposition 7.10 invoke Lemma 6.16, so the contractibility proof is incomplete as written. The gap looks repairable, but the repair is not in the paper. Theorem C and Theorem D are mostly independent of this gap, and Theorem C is already covered by Corrigan.

Who this is for. Geometric group theorists working on RAAG automorphism groups, cohomological dimension, and ell^2 invariants. It deserves a serious referee: the construction and the program are solid, and the flaw is local and plausibly fixable. I would send it to review, but I would not cite Theorem B until the Existence Lemma is repaired.

Referee Report

3 major / 5 minor

Summary. The paper generalizes the McCullough-Miller complex from free groups to arbitrary right-angled Artin groups. It introduces Γ-valid based partitions, the Γ-Whitehead poset Wh_Γ, and the complex MM_Γ as the simplicial realization of marked vertex types modulo the relation generated by carried automorphisms. Theorem A states that ΣPOut(A_Γ) acts on MM_Γ with free abelian stabilizers and fundamental domain |Wh_Γ|. The bulk of the paper (Sections 5–7) develops compatibility, refinement, disjunction, and reductivity, and uses Day's peak-reduction theorem to prove the main contractibility result (Theorem B). Section 8 applies the action to compute the cohomological dimension and the ℓ²-Betti numbers of ΣPOut(A_Γ).

Significance. If Theorem B is established, the paper provides a Deligne-type contractible complex for ΣPOut(A_Γ) with free abelian cell stabilizers, a genuine extension of the free-group McCullough-Miller construction. The stabilizer computation in Lemma 5.3 and the translation of the combinatorial framework to RAAGs are carried out in detail, and Example 6.13 correctly identifies a defect in the original McCullough-Miller Lemma 3.8. The paper also credits concurrent work by Corrigan and Abgrall for Theorem C, so the main novelty is the complex itself and its contractibility proof.

major comments (3)

[Section 6.1, Lemma 6.16] The displayed chain red_W(α_1^{-1}) = ∥u0∥_W − ∥α_1^{-1}(u0)∥_W = ∥u0∥_{W'} − ∥α_1(u0)∥_{W'} > 0 is not justified. For a partial conjugation α_1 = C_a^A, the W'-reductivity satisfies red_{W'}(α_1^{-1}) = red_W(α_1^{-1}) − 2|A|, because each generator b^k with b∈A is sent to a^{-1} b^k a and gains two letters. The equality with ∥u0∥_{W'} − ∥α_1(u0)∥_{W'} is therefore false as written, and it also shifts from α_1^{-1} to α_1. The strict inequality red_{W'}(α_1^{-1}) > 0 that the peak-reduction argument actually yields does imply red_W(α_1^{-1}) > 0, but only through the missing comparison red_{W'}(α_1^{-1}) = red_W(α_1^{-1}) − 2|A|; this step must be supplied.
[Section 6.1, Lemma 6.16] The argument ruling out a non-partial first factor is circular as written. The proof defines W' = W ∪ {b^k | b∈VΓ} before peak reduction, and then, after obtaining α_1, invokes a choice k ≥ ∥v∥_{W'}, with v never defined. Because W' itself contains b^k, the quantity ∥v∥_{W'} depends on the same k that is being chosen; choosing k after the factorization is produced is not legitimate, since the peak-reduction factorization depends on k. A repair must fix k in advance, or use a bound independent of k, and then rule out a non-partial α_1.
[Section 7, Lemma 7.9 and Proposition 7.10] The contractibility proof of Theorem B depends directly on Lemma 6.16. Lemma 7.9 invokes Lemma 6.16 to produce a strictly reductive partial conjugation P, and Proposition 7.10 uses Lemma 7.9 to prove contractibility of T. Consequently the gap in Lemma 6.16 is load-bearing: Theorem B is not established by the current text. If the repairs indicated in the two preceding comments are made, the remaining structure of the proof appears coherent.

minor comments (5)

[Section 8.1] The first sentence says 'In this subsection we prove Theorem D', but the subsection proves Theorem C; this should be corrected.
[Theorem D] In the statement of Theorem D, the first displayed equality has an unbalanced parenthesis: dimNG(Hi(MMΓ;N (ΣPOut(AΓ)) needs a closing parenthesis before the equals sign.
[Section 5, proof of Theorem A] In the proof of Theorem A, the phrase 'For the statement about centralizers' should read 'For the statement about stabilizers'.
[Section 7.2] The sentence 'The complex MMΓ is the union of the starts of the nuclear vertices' should read 'the union of the stars of the nuclear vertices'.
[Lemma 5.4] In the displayed calculation for αβ(c), the expression appears to interchange the roles of a and b: the definition gives the value b a c a^{-1} b^{-1} rather than a b c b^{-1} a^{-1} (depending on the convention for composing automorphisms). Please check and correct this display.

Circularity Check

1 steps flagged · score 3.0 of 10

Local self-referential choice of k in Lemma 6.16; the overall derivation is otherwise independent, with no self-citation or fitted-input circularity.

self definitional [Section 6.1, Lemma 6.16 (Existence Lemma), proof, last paragraph]
"W′ =W∪{bk|b∈VΓ} for some k>0. ... If α1 is not a partial conjugation, then for some a∈L and b∈VΓ, α1(b) is either ab or ba so |α1(bk)|VΓ−|bk|VΓ≥k and this means that if we chose k≥∥v∥W′ we get a contradiction."

The proof tries to choose k after W′ has already been defined as W ∪ {b^k : b ∈ VΓ}. The proposed contradiction requires k ≥ ∥v∥_{W′}, but ∥v∥_{W′} includes the k n length contributed by the added b^k words, so the condition is of the form k ≥ C + k n (up to the other terms of W). Such a k cannot be chosen; the definition of k is self-referential because the bound depends on the very set W′ that is defined using k. The intended contradiction—that a non-partial first factor adds at least k to the total length and therefore cannot occur—does not follow. This is load-bearing: Lemma 6.16 is invoked in Lemma 7.9 and Proposition 7.10 in the proof of Theorem B.

full rationale

The paper is not circular in the usual sense: Theorem A and Theorem B are built from external results (Day [10], Laurence [17], Toinet [24], McCullough–Miller [19], Quillen [20]), and Theorem C is honestly attributed also to Corrigan [7]. There is no self-citation that carries the argument, no fitted parameter renamed as a prediction, and no known result merely renamed as a new structure. The only circular feature I can exhibit is the self-referential choice of k in Lemma 6.16: W′ is defined using k, and then the proof requires k ≥ ∥v∥_{W′}, a quantity that depends on W′ and hence on k. Since ∥v∥_{W′} grows with k (the added words b^k contribute k n to the height), the required inequality cannot be satisfied for large k, so the contradiction is not obtained. This invalidates the proof of Lemma 6.16 as written, and because Lemma 7.9 and Proposition 7.10 depend on it, the contractibility proof of Theorem B is incomplete as written. There is also a separate, non-circular correctness gap in the displayed equality red_W(α_1^{-1}) = ∥u0∥_{W′} − ∥α_1(u0)∥_{W′}, which shifts from α_1^{-1} to α_1 and treats the added b^k words as having uniform length change; this is an unjustified computation rather than a circular derivation. Because the central theorem does not reduce to its inputs by construction, but one load-bearing technical step contains a genuine self-referential choice, the appropriate score is 3 rather than 0 or 6.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central construction introduces no fitted constants or ad hoc parameters. It depends on standard external results in RAAG automorphism theory and poset topology; the most load-bearing external input is Day's peak reduction theorem. The suspected gap in Lemma 6.16 is an issue of how the paper applies that theorem, not a circular dependence.

assumptions (5)

domain assumption Laurence's theorem that partial conjugations generate SigmaPAut(A_Gamma) (Theorem 3.4, citing [17])
Used throughout to describe carried automorphisms and stabilize rank computations; accepted external result.
domain assumption Toinet/Koban-Piggott presentation of SigmaPAut(A_Gamma) in terms of partial conjugations and commutation relations for shared/subordinate components (Theorem 3.6, citing [16])
Basis for Lemmas 5.1 and 5.2 on commuting outer automorphisms; external result.
domain assumption Day's peak-reduction theorem for long-range Whitehead automorphisms of RAAGs (Theorem B of [10])
Used in Lemma 6.16 to find a strictly reductive factor; external, but the paper's application of it is the location of the suspected gap.
domain assumption McCullough-Miller contractibility and stabilizer theorems for free groups ([19], Theorem 2.6 and Lemma 2.7)
The template the paper generalizes; invoked as background and for the original construction.
standard math Quillen's poset lemmas (Lemmas 7.3, 7.4, 7.5), Serre's cohomological dimension bound (Proposition 8.1), and McCammond-Meier l2 theorem (Theorem 8.4)
Standard tools used in the contractibility and application sections.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The McCullough-Miller complex for right angled Artin groups." pith.science (2026). https://pith.science/paper/NJQ3LKYM

@misc{pith2026250603377,
  author       = {Pith},
  title        = {Pith review of: The McCullough-Miller complex for right angled Artin groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NJQ3LKYM}},
  note         = {Machine review of arXiv:2506.03377}
}

abstract

McCullough and Miller constructed a contractible complex on which the pure symmetric automorphism group of a free group acts with free abelian stabilizers. This complex has been used for computations such as the cohomological dimension of these groups, their cohomology rings, and results about $\ell^2$-Betti numbers or BNRS-invariants. We generalize this construction to pure symmetric automorphism groups of arbitrary RAAGs and exhibit applications of this generalization.

Figures

Figures reproduced from arXiv: 2506.03377 by the authors.

**Figure 2.** A flower diagram For example, for the vertex type B of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Two based partitions with a crossing Definition 2.8 (Crossings and disjoint based partitions). Let Ai , Bj be two based partitions with operative factors i and j respectively. Two petals P ∈ Ai and Q ∈ Bj cross if i ̸= j, P ∩ Q ̸= ∅, i /∈ Q and j /∈ P. The number of crosses between two based partitions is the number of such pairs P and Q and is denoted cr(Ai , Bj ). We say that two partitions are disjoint if cr(Ai ,… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Graph components 4. Construction of the Γ-McCullough Miller space We begin by defining the analogous to the notion of based partitions in our case. In the case where Γ has no edges, so AΓ is free, we can identify possible partial conjugations by a vertex a with petals …

**Figure 5.** Figure 5: A graph Γ such that the based partitions of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Refinement and disjunction of A3 respect to B5 Definition 5.8 (Disjunction). Let Aa and Bb be based partitions with operative factors a, b such that a /∈ stΓ(b). Let A ′ a be the refinement of Aa with respect to Bb . We define the disjunction A ′′ a of Aa with respect …

**Figure 7.** Figure 7: Crossings between Sc , Aa and Ax - If a /∈ H1, we get (2) with H = H1. □ Remark. Observe that a consequence of the proof of item ii) in Lemma 5.15 is that if Ax , Aa are based partitions of a vertex type A such that both cross a based partition Sc and Ax is strictly Sc…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

25 extracted references · 19 canonical work pages

[1]

Relative Untwisted Outer Space for Right-Angled Artin Groups

A. Abgrall,Relative untwisted outer Space for right-angled Artin groups. Preprint, https://arxiv.org/abs/2503.09588

work page Pith review arXiv
[2]

R. Boyd, C. Bregman,Embedding spaces of split links. Preprint, https://arxiv.org/abs/2207.00619

arXiv
[3]

Brady, J

N. Brady, J. McCammond, J. Meier, A. Miller,The Pure Symmetric Automorphisms of a Free Group Form a Duality Group, J. Algebra. 246, Issue 2, 881-896, 2001

work page 2001
[4]

BrownCohomology of Groups, Graduate Texts in Mathematics, Springer, 1982

K. BrownCohomology of Groups, Graduate Texts in Mathematics, Springer, 1982

work page 1982
[5]

D. J. Collins,Cohomological dimension and symmetric automorphisms of a free group, Commentarii Mathematici Helvetici, 64(1), 44-61, 1989

work page 1989
[6]

D. J. Collins, H. Zieschang,Rescuing the Whitehead method for free products I: Peak reduction, Mathematische Zeitschrift. 185, 487-504, 1984

work page 1984
[7]

Outer space and finiteness properties for symmetric automorphisms of RAAGs, and generalisations

G. Corrigan, Outer space and finiteness properties for symmetric automorphisms of RAAGs, and generalisations , https://arxiv.org/abs/2503.05527 33

work page Pith review arXiv
[8]

Davis, T

M.W. Davis, T. Januszkiewicz, I.J Leary,The l2-cohomology of hyperplane complements, Groups Geom. Dyn. 1, 301–309, 2007

work page 2007

Show all 25 references

[9]

Davis, I.J Leary,The l2-cohomology of Artin groups, J

M.W. Davis, I.J Leary,The l2-cohomology of Artin groups, J. London Math. Soc. (2) 68, 493-510, 2003

2003
[10]

M. B. Day,Peak reduction and finite presentations for automorphism group of right-angled Artin groups, Geometry & Topology, 13(2), 817-855, 2009

2009
[11]

M. B. Day, R. D. Wade,Subspace arrangements, BNS invariants and pure symmetric outer automorphisms of right-angled Artin groups, Groups, Geometry and Dynamics, 18, 173-206, 2015

2015
[12]

M. B. Day, R. D. Wade,Calculating the virtual cohomological dimension of the automorphism group of a RAAG, Bulletin of the London Mathematical Society 53.1, 259-273, 2021

2021
[13]

Ershov, M

M. Ershov, M. C. B. Zaremsky,Dense and empty BNSR-invariants of the McCool groupshttps://arxiv.org/abs/2505.18826
[14]

C. A. Jensen, J. McCammond, J. Meier,The integral cohomology of the group of loops. Geometry & Topology, 10(2), 759-784, 2006

2006
[15]

Groups Acting on Fractals, Hyperbolicity and Self-Similarity

A. Genevois,Mini-course on the embedding problem between RAAGs. Notes of a mini-course given at the CIRM in April 2022 for the thematic semester "Groups Acting on Fractals, Hyperbolicity and Self-Similarity" of the Institut Henri Poincaré. https://arxiv.org/abs/2304.05267

2022 arXiv
[16]

Koban, A

N. Koban, A. Piggott,The Bieri-Neumann-Strebel invariant of the pure symmetric automorphisms of a right-angled Artin group. Illinois J. Math. 58, 2014

2014
[17]

M. R. Laurence,A generating set for the automorphism group of a graph group, J. London Math. Soc. (2), 52(2): 318-334, 1995

1995
[18]

McCammond and J

J. McCammond and J. Meier, The hypertree poset and theℓ2-Betti numbers of the motion group of the trivial link. Mathematische Annalen, 328(4), 633-, 2004

2004
[19]

McCullough, A

D. McCullough, A. Miller,Symmetric Automorphisms of Free Products, Memoirs of the AMS. 582, 1996

1996
[20]

Quillen,Homotopy properties of the poset of nontrivialp-subgroups of a group, Adv

D. Quillen,Homotopy properties of the poset of nontrivialp-subgroups of a group, Adv. in Math, 28, 101-128, 1978

1978
[21]

Piggott,The symmetries of McCullough-Miller space, Algebra and discrete mathematics, 2012

A. Piggott,The symmetries of McCullough-Miller space, Algebra and discrete mathematics, 2012

2012
[22]

Serre,Cohomologie des groupes discrets, Annals of Math

J-P. Serre,Cohomologie des groupes discrets, Annals of Math. Studies 60, 77-169, 1971

1971
[23]

Servatius,Automorphisms of graph groups, J

H. Servatius,Automorphisms of graph groups, J. Algebra 126, 1989

1989
[24]

Toinet,A finitely presented subgroup of the automorphism group of a right-angled Artin group

E. Toinet,A finitely presented subgroup of the automorphism group of a right-angled Artin group. J. Group Theory 15, 2012

2012
[25]

Wiedmer, Right-angled artin groups as finite-index subgroups of their outer automorphism groups

M. Wiedmer, Right-angled artin groups as finite-index subgroups of their outer automorphism groups. Bulletin of the London Mathematical Society: Volume 56, Issue 3, 945-958, 2024. 34

2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Relative Untwisted Outer Space for Right-Angled Artin Groups

A. Abgrall,Relative untwisted outer Space for right-angled Artin groups. Preprint, https://arxiv.org/abs/2503.09588

work page Pith review arXiv

[2] [2]

R. Boyd, C. Bregman,Embedding spaces of split links. Preprint, https://arxiv.org/abs/2207.00619

arXiv

[3] [3]

Brady, J

N. Brady, J. McCammond, J. Meier, A. Miller,The Pure Symmetric Automorphisms of a Free Group Form a Duality Group, J. Algebra. 246, Issue 2, 881-896, 2001

work page 2001

[4] [4]

BrownCohomology of Groups, Graduate Texts in Mathematics, Springer, 1982

K. BrownCohomology of Groups, Graduate Texts in Mathematics, Springer, 1982

work page 1982

[5] [5]

D. J. Collins,Cohomological dimension and symmetric automorphisms of a free group, Commentarii Mathematici Helvetici, 64(1), 44-61, 1989

work page 1989

[6] [6]

D. J. Collins, H. Zieschang,Rescuing the Whitehead method for free products I: Peak reduction, Mathematische Zeitschrift. 185, 487-504, 1984

work page 1984

[7] [7]

Outer space and finiteness properties for symmetric automorphisms of RAAGs, and generalisations

G. Corrigan, Outer space and finiteness properties for symmetric automorphisms of RAAGs, and generalisations , https://arxiv.org/abs/2503.05527 33

work page Pith review arXiv

[8] [8]

Davis, T

M.W. Davis, T. Januszkiewicz, I.J Leary,The l2-cohomology of hyperplane complements, Groups Geom. Dyn. 1, 301–309, 2007

work page 2007