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Subgroups of right-angled Coxeter groups via Stallings-like techniques

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Pith's one-line read The paper proves that a subgroup of a right-angled Coxeter group is quasiconvex exactly when its standard completion—an edge-labeled cube complex built from the subgroup's generators—is finite, and uses this equivalence to detect…

desk verdict A genuinely useful Stallings-style completion framework for RACGs; the quasiconvexity characterization is solid, but the finite-index embeddability algorithm leans on a long hand-audited word-length bound. read the letter →

arxiv 1908.09046 v3 pith:3Z7NOU27 submitted 2019-08-23 math.GT math.GR

Pallavi Dani , Ivan Levcovitz This is my paper · ORCID
classification math.GTmath.GR MSC 20F6557M0720F55
keywords right-angledCoxetergroupsquasiconvexsubgroupsStallings-liketechniquescubecomplexesreflectionfinite-indexalgorithmicgrouptheoryseparability
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 introduces 'completions' for subgroups of right-angled Coxeter groups (RACGs): cube complexes built from a finite generating set by Stallings-style folds, cube attachments, and cube identifications. Its central theorem states that a subgroup is quasiconvex if and only if it is finitely generated and every standard completion is finite. The same complexes encode finite index, torsion, and normality, and they turn those properties into algorithmically checkable conditions. The authors use this to prove that reflection subgroups and one-ended Coxeter subgroups of 2-dimensional RACGs are quasiconvex, to give an algorithm deciding finite-index embeddability between RACGs, and to re-prove Haglund's separability theorem.

What carries the argument

The completion is the central object. Starting from a 'rose' graph whose petals are labeled by the chosen generator words, one repeatedly folds pairs of edges with the same label, attaches cubes whenever a tuple of incident edges has labels forming a clique in the defining graph, and identifies cubes with identical boundaries; the process stops when the complex is folded and cube-full. Being cube-full guarantees that no commuting relations are missing, so reduced words behave like geodesics in the complex; this is what lets finiteness of the completion control quasiconvexity and lets non-positively curved completions support the geometric arguments.

What would settle it

Find a triangle-free, non-almost-star graph Γ and a finite set of reflections R generating a finite-index subgroup of WΓ such that every trimmed generating set contains a word longer than the constant M(|V(Γ)|,|R|) supplied by Proposition 12.1; equivalently, exhibit a pair (Γ,Γ′) for which the algorithm of Theorem 12.8 says 'no' although WΓ′ embeds as a finite-index subgroup of WΓ.

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Extended reading notes

Core claim

For any finitely generated subgroup G of a RACG WΓ, the paper constructs a completion Ω: a folded, cube-full, edge-labeled cube complex whose loops based at the basepoint carry exactly the reduced words representing elements of G, and in which every reduced word for a group element labels a loop. Theorem 8.4 states that G is quasiconvex in WΓ if and only if G is finitely generated and some (equivalently every) standard completion is finite. A completion is finite exactly when it can be built in finite time, so quasiconvexity becomes decidable. The paper further shows that a completion determines whether G is finite-index (the complex is finite and every vertex is incident to an edge of every label), torsion-free (no loop reduces into a finite special subgroup), and normal (conditions on the core graph, with a computable reformulation).

Load-bearing premise

The finite-index embeddability algorithm rests on the bound in Proposition 12.1: for a triangle-free defining graph that is not almost star, every trimmed reflection generating set of a finite-index subgroup has word lengths bounded by a constant depending only on the graph and the number of reflections, so a subgroup needing a longer generator would escape the algorithm's enumeration.

Editorial extensions

If this is right

  • Quasiconvexity of a subgroup of a RACG is algorithmically detectable: build a standard completion and check whether it is finite.
  • For a quasiconvex subgroup given by finitely many words, there are algorithms to test torsion-freeness, compute its index, test whether a power of a given element lies in the subgroup, and test normality.
  • Every finitely generated reflection subgroup of a RACG is quasiconvex, because it admits a finite completion.
  • Every one-ended Coxeter subgroup of a 2-dimensional RACG is a reflection subgroup and hence quasiconvex.
  • There is an explicit algorithm that decides, given a 2-dimensional RACG WΓ and any RACG WΓ′, whether WΓ′ is isomorphic to a finite-index subgroup of WΓ, and outputs the embedding words when it is.

Reading between the lines

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

  • A natural next step, not taken in the paper, is to implement the standard completion algorithm on small defining graphs and compare the finite-completion criterion with known commensurability classifications; this would give empirical data on how sharp the length bound in Proposition 12.1 is.
  • Because the completion construction does not assume hyperbolicity, the same cube-full finiteness criterion may be adaptable to other graph products of groups whose defining graphs carry a similar commutativity structure, though the paper does not claim this.
  • The algorithm in Theorem 12.8 effectively reduces finite-index embeddability to a bounded search over trimmed reflection sets; if the bound in Proposition 12.1 can be made explicit and small, the theorem becomes a practical computational tool for commensurability questions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper develops a Stallings-style completion theory for subgroups of right-angled Coxeter groups (RACGs). For a finitely generated subgroup G < W_Γ, the authors construct a Γ-labeled cube complex, called a completion, via fold, cube-attachment, and cube-identification operations. They prove that properties of G are reflected in completions: quasiconvexity is equivalent to finiteness of a standard completion (Theorem A(1) / Theorem 8.4), finite index is characterized by finite full-valence resolved completions (Theorem 6.6), torsion is detected by loops with reduced labels in finite special subgroups (Proposition 4.6), and normality has a core-graph characterization (Theorem 5.3). The paper then applies this machinery to show that finitely generated reflection subgroups are quasiconvex (Theorem 10.5), that one-ended Coxeter subgroups of 2-dimensional RACGs are reflection subgroups and hence quasiconvex (Theorem 11.4 and Corollary 11.5), and to give an algorithm deciding finite-index embeddability between certain RACGs (Theorem 12.8). It also provides algorithms for several properties of quasiconvex subgroups (Theorem E) and gives new proofs of residual finiteness and of Haglund's separability theorem for quasiconvex subgroups.

Significance. If the results are correct, this is a substantial contribution to the geometric and algorithmic study of subgroups of RACGs. The completion construction is a genuinely new tool in this setting, and the characterization of quasiconvexity by finiteness of a completion is both conceptually clean and algorithmically useful. The paper contains many explicit, detailed arguments, and the main theorems are not obtained by circular reasoning: the completion is defined independently, and the axioms ledger is empty. The algorithmic applications, especially Theorems D and E, are strong and well-motivated. However, the paper's central algorithmic theorem depends on a long and only hand-audited combinatorial bound, and several statements rely on omitted proof details. These points need to be addressed before the paper can be accepted in its present form.

major comments (4)
  1. [Section 12, Proposition 12.1 and Lemmas 12.2–12.5] The completeness of the algorithm in Theorem 12.8 rests entirely on Proposition 12.1, because the algorithm enumerates M-admissible trimmed reflection sets and would silently miss a finite-index subgroup if the uniform length bound failed. The proof of Proposition 12.1 is a long chain of auxiliary lemmas, and I could not fully verify the crucial uniqueness claim in Lemma 12.4, namely that ``e_1 is the only edge of ``f(FT) dual to H_1``. In particular, the step where the maximal-prefix choice of the expressions w_i is used to rule out all other edges dual to H_1 deserves a more explicit and self-contained justification. Since a gap here would invalidate the main algorithmic theorem, I request that the proof of Proposition 12.1 be expanded, with special attention to Lemma 12.4 and to the maximal-prefix argument.
  2. [Section 5, Proposition 5.2] Proposition 5.2 is stated with the proof omitted, with the explanation that it follows closely from [KM02, Theorem 5.2]. This is not satisfactory, because in the present setting completions are not unique and need not even have the same homotopy type (Example 3.7), so uniqueness of the core graph is a nontrivial claim. The proposition is used in the proof of Theorem 5.3, which is one of the advertised characterizations in Theorem A(2). Please provide a complete proof, or at minimum a precise reduction to [KM02, Theorem 5.2] that accounts for the distinction between completions and Stallings graphs.
  3. [Section 3, Lemma 3.9] In the fold operation case, the proof says that path types 3 and 4 are handled similarly and omits them. These cases are not merely cosmetic: they cover loops based at the basepoint B that traverse the folded edge near the beginning or end of the loop, and they are needed for the iteration in Lemma 3.10, which in turn underpins Theorem 3.11. Please supply the missing arguments for types 3 and 4.
  4. [Introduction, Theorem D; Section 12, Theorem 12.8] The introduction states Theorem D as providing an algorithm that, given a one-ended 2-dimensional RACG W_Γ and any RACG W_Γ', decides whether W_Γ' embeds as a finite-index subgroup of W_Γ. However, the theorem actually proved in Section 12 assumes that Γ' has no isolated vertex. If the isolated-vertex case is genuinely excluded, the introductory and abstract statements need to be revised; if it is covered by a separate argument, that argument must be included. This mismatch concerns one of the paper's central advertised claims.
minor comments (5)
  1. [Section 5, Proposition 5.2] The statement contains a typo: the right-hand side reads ``C_2(Ω_1, B_2)``, but it should presumably be ``C_2(Ω_2, B_2)``.
  2. [Section 3, Example 3.7] The two completions are described as a torus and a Klein bottle obtained by attaching a square to the rose graph, but the attaching maps are not written out. Since the labels and the attaching words determine whether the complexes are Γ-labeled completions, the example would be much easier to check if the boundary words of the attached squares were specified explicitly.
  3. [Figures 2 and 7] Several figures omit edge labels or use very small labels, which makes the examples difficult to verify independently. Please add labels or explain the omitted labels in the captions.
  4. [Section 12, Theorem 12.8] In case (iv) of the proof, the reduction to the non-almost-star case via the kernel K' is clear in principle, but the sentence ``The theorem now follows`` hides the induction on the number of vertices. Since the algorithm is central, a short explicit statement of the induction and of why termination is guaranteed would improve readability.
  5. [Section 13, Lemma 13.1] The notation ``k^{(n mod 2)}`` is used without a prior definition. For clarity, state explicitly that this means the word k if n is odd and the empty word if n is even.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; main derivations are self-contained and do not reduce to their own inputs.

full rationale

The paper's central constructions are defined independently of the properties they characterize. A completion of a subgroup G is defined by abstract conditions (Definition 3.6), but existence is proved constructively: standard completions are built from a finite generating set via fold, cube-attachment, and cube-identification operations (Propositions 3.3 and 3.5, Theorem 3.11), with Lemmas 3.8-3.10 verifying that the resulting complex satisfies the defining properties. The quasiconvexity characterization (Theorem 8.4) is not a renaming: Lemma 8.1 compares distances in the completion with distances in the Cayley graph to G, Lemma 8.2 derives quasiconvexity from finiteness of any completion, and Lemma 8.3 derives finiteness of every standard completion from quasiconvexity using bounds on core-graph vertices and hyperplane finiteness. These are independent geometric arguments. The index characterization (Theorem 6.6), torsion characterization (Proposition 4.6), and normality characterization (Theorems 5.3 and Proposition 13.2) likewise each have proofs that do not assume the conclusion. The reflection-subgroup results use the Dyer-Deodhar theorem and standard Coxeter-group facts as external inputs, and the Coxeter-subgroup theorem (Theorem 11.4) is proved through Lemmas 11.1-11.3 without invoking quasiconvexity. In Section 12, Theorem 12.8 enumerates M-admissible trimmed reflection sets; Proposition 12.1 supplies the required bound under the hypothesis that the set generates a finite-index subgroup. This is a genuine correctness-sensitive step and its proof is hand-audited rather than machine-verified, but it is not circular: the enumeration is not defined as 'all finite-index subgroups' and the theorem would be incomplete, not tautological, if the bound failed. The self-reference to [DL20] is a forward pointer to a companion paper and is not load-bearing for any theorem proved here; the omitted proof of Proposition 5.2 follows an external argument of Kapovich-Miasnikov and is not used in the headline quasiconvexity theorem. The new proof of Haglund's separability result uses the independently built full-valence extension and external residual-finiteness facts, so it does not smuggle in the conclusion. Overall, no predicted quantity is fitted into the input and no load-bearing step reduces by construction to an earlier claim of the paper.

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

The central claim depends on standard theorems of Coxeter group theory and CAT(0) cube complexes, listed as axioms. No free parameters are fitted to data. No new entities in the natural-science sense are postulated; the completions are constructed objects whose existence is proven.

assumptions (6)
  • standard math Tits' solution to the word problem and deletion property for right-angled Coxeter groups
    Stated in Section 2.2 and used to move between expressions of the same group element in Lemmas 3.8, 4.1, 4.2, and 13.1.
  • standard math Every finite subgroup of a RACG is conjugate into a special finite subgroup (clique subgraph)
    Davis [Dav08, Theorem 12.3.4], invoked in Proposition 4.6 and Lemma 11.1 to analyze torsion and involutions.
  • standard math Reflection subgroups of Coxeter groups are Coxeter groups, and trimmed reflection sets are standard Coxeter generating sets
    Dyer [Dye90] and Deodhar [Deo89], used in Proposition 12.7 and Theorem 12.8 to recognize subgroups generated by reflections.
  • standard math A RACG is one-ended iff its defining graph is connected and has no separating clique
    Mihalik-Tschantz [MT09], stated in Section 2.1; used to translate one-endedness of Coxeter subgroups into absence of isolated vertices in Theorem 11.4.
  • standard math Graph products, and in particular RACGs, have unique defining graphs up to isomorphism
    Radcliffe [Rad03], used in Theorem 12.8 to decide whether a subgroup generated by reflections is isomorphic to a given RACG.
  • standard math Virtual retracts of residually finite groups are separable
    Haglund [Hag08, Proposition 3.8], used in Theorem 9.5 to derive separability of quasiconvex subgroups.

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Pith. "Pith review of Subgroups of right-angled Coxeter groups via Stallings-like techniques." pith.science (2026). https://pith.science/paper/3Z7NOU27

@misc{pith2026190809046,
  author       = {Pith},
  title        = {Pith review of: Subgroups of right-angled Coxeter groups via Stallings-like techniques},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3Z7NOU27}},
  note         = {Machine review of arXiv:1908.09046}
}
read the original abstract

We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional RACG. We provide an algorithm that determines whether a given one-ended, 2-dimensional RACG is isomorphic to some finite-index subgroup of another given RACG. In addition, we answer several algorithmic questions regarding quasiconvex subgroups. Finally, we give a new proof of Haglund's result that quasiconvex subgroups of RACGs are separable.

Figures

Figures reproduced from arXiv: 1908.09046 by the authors.

Figure 1
Figure 1. A completion Ω for the Γ1-labeled complex X. Example 3.2 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. A completion Ω′ for the Γ2-labeled complex X. Next we describe a sequence of operations to be performed to the finite folded complex Ωi+j . Choose a vertex v of Ωi+j . Consider a maximal tuple of edges incident to v such that their labels form a clique in Γ. If there is no cube in Ωi+j whose boundary contains the tuple of edges, then attach an appropriately labeled cube of the appropriate dimension along the tuple o… view at source ↗
Figure 3
Figure 3. The graph on the top shows a path of type 2 in Ω. The two graphs below it show the two possible choices of preimages for p. The graph￾loops ¯qi1 , . . . , q¯ik′ form a subsequence of the graph-loops ¯q1, . . . , q¯k consisting of those based at ¯v1. Similarly, ¯qik′ , . . . , q¯ik′′−1 are edges between ¯v1 and ¯v2 and q¯ik′′, . . . , q¯ik are graph-loops based at ¯v2. Note that some of the vertices and edges shown m… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: On the other hand, if i ′ − i = 1, then as Ω is folded, α either traverses an edge labeled by s twice in opposite directions, or a graph-loop labeled by s twice consecutively. In either case we can simply remove both occurrences of α from p to obtain a new path p1 with…
Figure 5
Figure 5. Figure 5: Figure illustrating Example 6.7 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: Figure illustrating Example 8.5. 9. Residual finiteness and separability In this section we give a proof using completions of the well-known result that RACGs are residually finite. We additionally give a new proof of a result of Haglund which states that quasiconvex s…
Figure 7
Figure 7. Figure 7: A completion for a reflection subgroup. For 2-dimensional RACGs, we obtain the following stronger result which shows that the time-complexity of the algorithm which builds the completion of a reflection subgroup is bounded by the size of words in the generating set of …

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Works this paper leans on

41 extracted references · 40 canonical work pages

  1. [1]

    G. N. Arzhantseva and P.-A. Cherix, On the C ayley graph of a generic finitely presented group , Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 4, 589--601. 2115727

    work page 2004

  2. [2]

    Ian Agol, The virtual H aken conjecture , Doc. Math. 18 (2013), 1045--1087, With an appendix by Agol, Daniel Groves, and Jason Manning

    work page 2013

  3. [3]

    G. N. Arzhantseva and A. Yu. Ol'shanskii, Generality of the class of groups in which subgroups with a lesser number of generators are free, Mat. Zametki 59 (1996), no. 4, 489--496, 638. 1445193

    work page 1996

  4. [4]

    G. N. Arzhantseva, Generic properties of finitely presented groups, PhD thesis, Moscow Lomonosov State University, 1998

    work page 1998

  5. [5]

    , On groups in which subgroups with a fixed number of generators are free, Fundam. Prikl. Mat. 3 (1997), no. 3, 675--683. 1794135

    work page 1997

  6. [6]

    Algebra 26 (1998), no

    , Generic properties of finitely presented groups and H owson's theorem , Comm. Algebra 26 (1998), no. 11, 3783--3792. 1647075

    work page 1998

  7. [7]

    , A property of subgroups of infinite index in a free group, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3205--3210. 1694447

    work page 2000

  8. [8]

    Patrick Bahls, The isomorphism problem in C oxeter groups , Imperial College Press, London, 2005

    work page 2005

Show all 41 references
  1. [9]

    32, Springer, 2005

    Anders Bjorner and Francesco Brenti, Combinatorics of C oxeter groups , Graduate Texts in Mathematics, vol. 32, Springer, 2005

    2005

  2. [10]

    Howlett, A finiteness property and an automatic structure for C oxeter groups , Math

    Brigitte Brink and Robert B. Howlett, A finiteness property and an automatic structure for C oxeter groups , Math. Ann. 296 (1993), no. 1, 179--190

    1993

  3. [11]

    Bridson and Andr \'e Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol

    Martin R. Bridson and Andr \'e Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999

    1999

  4. [12]

    Benjamin Beeker and Nir Lazarovich, Stallings' folds for cube complexes, Israel J. Math. 227 (2018), no. 1, 331--363

    2018

  5. [13]

    Samuel Brown, Geometric structures on negatively curved groups and their subgroups, P h D thesis , University College London, 2016

    2016

  6. [14]

    John Crisp and Luisa Paoluzzi, Commensurability classification of a family of right-angled C oxeter groups , Proc. Amer. Math. Soc. 136 (2008), no. 7, 2343--2349

    2008

  7. [15]

    Montserrat Casals-Ruiz, Embeddability and universal theory of partially commutative groups, Int. Math. Res. Not. IMRN (2015), no. 24, 13575--13622

    2015

  8. [16]

    Pierre-Emmanuel Caprace and Michah Sageev, Rank rigidity for CAT (0) cube complexes , Geom. Funct. Anal. 21 (2011), no. 4, 851--891

    2011

  9. [17]

    V, International Press and Higher Education Press, 2018

    Pallavi Dani, The large-scale geometry of right-angled C oxeter groups , Handbook of Group Actions, vol. V, International Press and Higher Education Press, 2018

    2018

  10. [18]

    Davis, The geometry and topology of C oxeter groups , London Mathematical Society Monographs Series, vol

    Michael W. Davis, The geometry and topology of C oxeter groups , London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008

    2008

  11. [19]

    Deodhar, A note on subgroups generated by reflections in C oxeter groups , Arch

    Vinay V. Deodhar, A note on subgroups generated by reflections in C oxeter groups , Arch. Math. (Basel) 53 (1989), no. 6, 543--546

    1989

  12. [20]

    Davis and Tadeusz Januszkiewicz, Right-angled A rtin groups are commensurable with right-angled C oxeter groups , J

    Michael W. Davis and Tadeusz Januszkiewicz, Right-angled A rtin groups are commensurable with right-angled C oxeter groups , J. Pure Appl. Algebra 153 (2000), no. 3, 229--235

    2000

  13. [21]

    Pallavi Dani and Ivan Levcovitz, Right-angled A rtin subgroups of right-angled C oxeter and A rtin groups , arXiv:2003.05531, 2020

    2003 arXiv

  14. [22]

    Pallavi Dani, Emily Stark, and Anne Thomas, Commensurability for certain right-angled C oxeter groups and geometric amalgams of free groups , Groups Geom. Dyn. 12 (2018), no. 4, 1273--1341

    2018

  15. [23]

    Algebra 135 (1990), no

    Matthew Dyer, Reflection subgroups of C oxeter systems , J. Algebra 135 (1990), no. 1, 57--73

    1990

  16. [24]

    Gruber, Infinitely presented graphical small cancellation groups, PhD thesis, University of Vienna, 2015, http://othes.univie.ac.at/38520/

    Dominik. Gruber, Infinitely presented graphical small cancellation groups, PhD thesis, University of Vienna, 2015, http://othes.univie.ac.at/38520/

    2015

  17. [25]

    Dedicata 135 (2008), 167--209

    Fr\' e d\' e ric Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008), 167--209

    2008

  18. [26]

    Christopher Hruska, Emily Stark, and Hung Cong Tran, Surface group amalgams that (don't) act on 3-manifolds, arXiv:1705.01361, to appear in American Journal of Mathematics, 2017

    G. Christopher Hruska, Emily Stark, and Hung Cong Tran, Surface group amalgams that (don't) act on 3-manifolds, arXiv:1705.01361, to appear in American Journal of Mathematics, 2017

    2017 arXiv

  19. [27]

    Wise, Special cube complexes, Geom

    Fr\' e d\' e ric Haglund and Daniel T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551--1620

    2008

  20. [28]

    , Coxeter groups are virtually special, Adv. Math. 224 (2010), no. 5, 1890--1903

    2010

  21. [29]

    Sang-hyun Kim and Thomas Koberda, Embedability between right-angled A rtin groups , Geom. Topol. 17 (2013), no. 1, 493--530

    2013

  22. [30]

    Algebra 248 (2002), no

    Ilya Kapovich and Alexei Myasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248 (2002), no. 2, 608--668

    2002

  23. [31]

    Algebra 488 (2017), 442--483

    Olga Kharlampovich, Alexei Miasnikov, and Pascal Weil, Stallings graphs for quasi-convex subgroups, J. Algebra 488 (2017), 442--483

    2017

  24. [32]

    Michael Mihalik and Steven Tschantz, Visual decompositions of C oxeter groups , Groups Geom. Dyn. 3 (2009), no. 1, 173--198

    2009

  25. [33]

    J. P. McCammond and D. T. Wise, Coherence, local quasiconvexity, and the perimeter of 2-complexes, Geom. Funct. Anal. 15 (2005), no. 4, 859--927

    2005

  26. [34]

    Radcliffe, Rigidity of graph products of groups, Algebr

    David G. Radcliffe, Rigidity of graph products of groups, Algebr. Geom. Topol. 3 (2003), 1079--1088

    2003

  27. [35]

    Schupp, Coxeter groups, 2-completion, perimeter reduction and subgroup separability, Geom

    Paul E. Schupp, Coxeter groups, 2-completion, perimeter reduction and subgroup separability, Geom. Dedicata 96 (2003), 179--198

    2003

  28. [36]

    Stallings, Topology of finite graphs, Invent

    John R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551--565

    1983

  29. [37]

    Algebra 438 (2015), 337--378

    Markus Steenbock, Rips- S egev torsion-free groups without the unique product property , J. Algebra 438 (2015), 337--378. 3353035

    2015

  30. [38]

    Wise, Cores for quasiconvex actions, Proc

    Michah Sageev and Daniel T. Wise, Cores for quasiconvex actions, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2731--2741

    2015

  31. [39]

    B. A. F. Wehrfritz, Infinite linear groups, Queen Mary College Mathematical Notes, Queen Mary College, Department of Pure Mathematics, London, 1969

    1969

  32. [40]

    Wise, The structure of groups with a quasiconvex hierarchy, available at https://www.math.mcgill.ca/wise/papers.html, 2011

    Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, available at https://www.math.mcgill.ca/wise/papers.html, 2011

    2011

  33. [41]

    117, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2012

    , From riches to raags: 3-manifolds, right-angled A rtin groups, and cubical geometry , CBMS Regional Conference Series in Mathematics, vol. 117, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2012

    2012

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