Pith. sign in

REVIEW 2 major objections 3 minor 25 references

The braid group Bn is not a quotient of a quasi-Coxeter interval group of type Dn

T0 review · 2 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

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

Pith's one-line read For n≥5, no proper quasi-Coxeter interval group of type D_n admits a surjective homomorphism onto the braid group B_n.

desk verdict Theorem 1.1 is a real new result with a clean MCG proof; the only substantive issue is an under-explained braid relation in Claim 1 that should be made explicit, plus minor typos. read the letter →

arxiv 2607.14369 v1 pith:4IZPINHJ submitted 2026-07-15 math.GR

Barbara Baumeister , Luis Paris , Sarah Rees This is my paper · ORCID
classification math.GR MSC 20F36
keywords intervalgroupsquasi-CoxeterelementsbraidArtinCoxetertypeDmappingclassendomorphismsofGarside
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that, for n≥5, none of the interval groups associated with proper quasi-Coxeter elements of the Coxeter group of type D_n can be mapped onto the braid group B_n on n strands. This is stronger than merely knowing these groups are not isomorphic to the Artin group of type D_n, because the ordinary Artin group does have B_n as a quotient. The result rules out any decomposition of those interval groups that would give a surjection onto the braid group. The proof is new in this area: it uses the classification of endomorphisms of braid groups and mapping class group computations. A large part of the paper is an expository account of the mapping class group tools needed.

What carries the argument

The load-bearing input is a rigidity theorem for endomorphisms of the braid group B_n (n≥5): every endomorphism is either cyclic or conjugate to one of the maps t_i ↦ t_i^ε Δ^{2k} with ε=±1 and k∈Z. The interval group G([1,w]) is presented as Q_{n,p}, with two embeddings κ, κ' of B_n that let a candidate surjection φ be converted into two endomorphisms of B_n. Mapping class group arguments—Dehn twists, half-twists, intersection numbers, and centralizers of partial generating sets in the punctured disc—are then used to pin down the conjugating element and extract a contradiction.

What would settle it

Check whether the relation s'_p = s_{p−1} s'_p s_{p−1} s'^{-1}_p s^{-1}_{p−1} holds in the group Q_{n,p}; if not, the proof of Claim 1 is invalid. More directly, search for a surjective homomorphism from Q_{5,2} onto B_5; a single such surjection falsifies the theorem.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is that for n≥5 and any proper quasi-Coxeter element w of the Coxeter group W[D_n], the interval group G([1,w]) admits no surjective homomorphism onto the braid group B_n. Since A[D_n] does map onto B_n, this yields as a corollary a new proof that these interval groups are not isomorphic to A[D_n]. The proof assumes a surjection φ from a presentation Q_{n,p} of G([1,w]) to B_n and constructs two endomorphisms of B_n by composing φ with two natural embeddings of B_n into Q_{n,p}. Using the classification of endomorphisms of B_n, it shows both are conjugate to the same standard map t_i ↦ t_i Δ^{2k}, then forces k=0. Combining centralizer computations in the mapp

Load-bearing premise

The proof of Claim 1 assumes the identity φ(s'_p)=φ(s_{p−1} s'_p s_{p−1} s'^{-1}_p s^{-1}_{p−1}) holds in Q_{n,p}, but this identity is not derived from the defining presentation; if it does not hold, the argument that ψ and ψ' are non-cyclic fails, and the rigidity theorem cannot be invoked.

Editorial extensions

If this is right

  • If true, no proper quasi-Coxeter interval group of type D_n (n≥5) can have B_n as a quotient, blocking any Artin-group-style semidirect product decomposition of these groups.
  • The result gives a second, independent route to the known non-isomorphism between these interval groups and the Artin group of type D_n.
  • The proof introduces mapping class group rigidity arguments into the study of interval groups, a technique that may transfer to other Coxeter types with proper quasi-Coxeter elements.
  • The statements are proved for n≥5; the case n=4 remains open under this method, though the non-isomorphism corollary already holds for n≥4.

Reading between the lines

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

  • A reader might expect that the same rigidity argument rules out not only surjections but any homomorphism from these interval groups to B_n whose image is not cyclic; the paper only claims surjectivity.
  • The unproved identity in the proof of Claim 1 is a potential soft spot; if it fails, the proof could be repaired by finding a different way to rule out cyclicity, or the theorem itself might fail.
  • Since the only proper quasi-Coxeter elements in D_n appear for n≥4, extending the method to n=4 would require a substitute for the braid group endomorphism classification, which is not covered by the classification used here.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper proves Theorem 1.1: for n ≥ 5, an interval group G([1,w]) associated with a proper quasi-Coxeter element w of the Coxeter group W[D_n] admits no surjective homomorphism onto the braid group B_n. The proof uses the presentation of such interval groups as quotients Q_{n,p} of Artin groups of a graph Ω_{n,p} (Theorem 3.1, from [BNR23]) and the classification of endomorphisms of B_n by Castel and by Chen–Kordek–Margalit (Theorem 3.2). Assuming a surjection φ: Q_{n,p} → B_n, the authors define two homomorphisms ψ, ψ': B_n → B_n by composing φ with the two natural inclusions of the standard generators t_i into Q_{n,p} (t_p ↦ s_p or s'_p). They argue that ψ and ψ' are not cyclic; hence, by Theorem 3.2, they are conjugate to the standard endomorphisms α_{ε,k}. A comparison of abelianizations forces ε=μ and k=l, and then surjectivity forces k=l=0. The remaining steps use mapping class group computations of centralizers and intersection numbers to show that the images of s_p and s'_p must agree, finally contradicting the defining relation of Q_{n,p}. Section 2 provides a self-contained exposition of the mapping class group tools.

Significance. If the proof is correct, the result is a genuine strengthening of the known non-isomorphism between proper quasi-Coxeter interval groups of type D_n and the Artin group A[D_n] (Corollary 1.2). It rules out not only isomorphisms but all surjective homomorphisms onto B_n, which is a natural first step toward understanding the possible decompositions of these interval groups. The proof strategy is original and valuable: it combines the interval-group presentations of [BNR23] with deep results on endomorphisms of braid groups, and the expository Section 2 makes the topological machinery accessible to a group-theoretic audience. The intersection-number arguments in Claims 5 and 6 are concrete and checkable. The main gap is a missing justification of a key algebraic identity in Claim 1, which must be repaired before the argument is complete.

major comments (2)
  1. [Section 3, Claim 1] The displayed chain of equalities in the proof of Claim 1 is not derived from the stated presentation of Q_{n,p}. The first equality asserts s'_p = s_{p−1} s'_p s_{p−1} s'^{-1}_p s^{-1}_{p−1}, which holds only if s_{p−1} and s'_p satisfy the braid relation of length 3; the fourth equality uses that s_p and s'_p commute. Neither relation is explicitly stated in the text; they must be inferred from the graph Ω_{n,p} in Figure 3.1, whose edge set is not described in words and is not visible in the version of the manuscript I reviewed. This step is load-bearing: without excluding the cyclic case, Theorem 3.2 cannot be applied to ψ and ψ′. The authors should specify the graph Ω_{n,p} explicitly (or include a clearly labelled figure) and verify the two relations used in the chain, or replace the chain with a derivation from the presentation and the defining relation of Q_{n,p}.
  2. [Section 3, definition of Q_{n,p} and Claim 5] The proof of Claim 5 says 'Since s_p and s'_p commute', but this assertion is not a consequence of the presentation as written. It becomes true only if the graph Ω_{n,p} has no edge between s_p and s'_p (for instance, if they are opposite vertices of a 4-cycle with s_{p−1}—s'_p—s_{p+1} and s_{p−1}—s_p—s_{p+1}); this is consistent with Claim 1 only under the same graph hypothesis. The manuscript should state the adjacency relations of Ω_{n,p} explicitly, because multiple parts of the proof depend on the exact edge set: the existence of the two inclusions κ, κ′ (which requires both s_{p−1}—s_p—s_{p+1} and s_{p−1}—s'_p—s_{p+1} to be paths with braid relations) and the commutation of s_p with s'_p. Without such a statement, the proof is not verifiable from the text.
minor comments (3)
  1. [Section 3, Claim 1] The sentence 'We prove in the same way that ψ′ is not cyclic' is terse. Since the graph is not symmetric in s_p and s'_p in an obvious way, a short indication of the argument for ψ′ would help. The proof for ψ relies on the braid relation between s_{p−1} and s'_p; the proof for ψ′ likely relies on the analogous relation between s_{p−1} and s_p, but this should be stated.
  2. [Section 3, Claim 5, Case 2] In the inequality displayed after Theorem 2.10, the symbol 'b' appears (e.g. i(h([c]),[b])). It should be b_2, the regular boundary of e_2; the current notation is confusing.
  3. [Section 3, Claim 4] The centraliser computations are only sketched by reference to Lemma 2.15. While the cases are plausible and the proof is analogous, the reader would benefit from a few more details in the cases p=2 and p=n−2, where the centraliser includes a half-twist generator.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 is derived from independent presentation and rigidity results; no step reduces to its own conclusion.

full rationale

The derivation chain for Theorem 1.1 is: (i) replace G([1,w]) by the presented group Q_{n,p} via Theorem 3.1, citing [BNR23]; (ii) define two push-forwards psi, psi' of a hypothetical surjection phi; (iii) rule out the cyclic case in Claim 1; (iv) apply the Castel/CKM classification Theorem 3.2; (v) use abelianization, centralizer and intersection arguments to force phi(s'_p)=t_p^eps; (vi) contradict the defining relation of Q_{n,p}. No step assumes the conclusion that B_n is not a quotient. The load-bearing citations [BNR23] for the presentation of G([1,w]) and [Cas16]/[CKM19] for the classification of braid-group endomorphisms are independent published results with stated assumptions that do not include the target conclusion. The authors' own previous non-isomorphism theorem [BHNR23b] is cited only for Corollary 1.2, not used to prove Theorem 1.1. The apparent gap flagged in Claim 1—the equality phi(s'_p)=phi(s_{p-1}s'_p s_{p-1}s'^{-1}_p s^{-1}_{p-1})—is not an invented relation: it is the Coxeter braid relation for the edge s_{p-1}--s'_p in the graph Omega_{n,p} defining A[Omega_{n,p}] (Figure 3.1), and Q_{n,p} is its quotient; the later simplification uses the cyclic assumption h=phi(s_{p-1})=phi(s_p) and the commutativity of s_p and s'_p. This is a proof-detail issue, not a circular step. The Remark limiting the theorem to n>=5 is a scope limitation, not a circular step. There is no fitted or renamed quantity, no uniqueness theorem imported from the authors, and no ansatz smuggled in via self-citation; the result is self-contained given its cited external inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new constants, entities, or fitted parameters. Its central claim rests on external theorems—the BNR23 presentation of quasi-Coxeter interval groups and Castel/CKM rigidity of braid-group endomorphisms—plus standard mapping class group facts. The only questionable element is an unsupported equality in Claim 1 that the authors should either prove or correct.

assumptions (4)
  • domain assumption Theorem 3.1: For n≥4 and every proper quasi-Coxeter element w of W[D_n], G([1,w]) is isomorphic to Q_{n,p} for some p∈{2,...,n−2} (cited from [BNR23, Thm 5.1]).
    The proof of Theorem 1.1 replaces the interval group with Q_{n,p} throughout; this presentation theorem is taken from the authors' earlier paper and is not reproved here. If it is wrong, the theorem has no proof.
  • domain assumption Theorem 3.2: Every endomorphism of B_n (n≥5) is either cyclic or conjugate to α_{ε,k}: t_i ↦ t_i^ε Δ^{2k} (Castel [Cas16], Chen–Kordek–Margalit [CKM19]).
    The contradiction argument classifies ψ and ψ' using this rigidity result; the proof does not establish it.
  • standard math The mapping class group facts in Section 2: Artin's isomorphism B_n ≅ M(D,P_n), Dehn twist/half-twist relations, Epstein's minimal-position criterion, intersection-number estimates, and centralizer computations.
    These are standard results cited to [FM12], [Eps66], [FLP79]; the paper supplies exposition but not full proofs.
  • ad hoc to paper In Claim 1, the equality φ(s'_p)=φ(s_p) under cyclicity of ψ relies on a relation in Q_{n,p} that is not explicit in the stated presentation.
    No derivation or citation is given for this relation in the provided text; the proof depends on it to show ψ and ψ' are non-cyclic.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The braid group Bn is not a quotient of a quasi-Coxeter interval group of type Dn." pith.science (2026). https://pith.science/paper/4IZPINHJ

@misc{pith2026260714369,
  author       = {Pith},
  title        = {Pith review of: The braid group Bn is not a quotient of a quasi-Coxeter interval group of type Dn},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4IZPINHJ}},
  note         = {Machine review of arXiv:2607.14369}
}
abstract

We prove that, for $n \ge 5$, an interval group associated with a proper quasi-Coxeter element of the Coxeter group of type $D_n$ admits no surjective homomorphism onto the braid group on $n$ strands. In particular, this provides an alternative proof that such a group is not isomorphic to the Artin group of type $D_n$. The proof relies on techniques from the theory of mapping class groups, and a large part of the paper provides an exposition of this theory for non-specialists.

Figures

Figures reproduced from arXiv: 2607.14369 by the authors.

Figure 2.1
Figure 2.1. Standard Dehn twist. A simple closed curve in (Σ, P) is an embedding a : S 1 → Σ whose image does not intersect either 𝜕Σ or P. We say that a is essential if its image does not bound any disc embedded in Σ that contains 0 or 1 elements of P. Let a : S 1 → Σ be a simple and essential closed curve. We choose an embedding aˆ : A → Σ whose image does not intersect either 𝜕Σ or P, and such that aˆ( 1 2 ,z) = a(z) for eac… view at source ↗
Figure 2.2
Figure 2.2. Dehn twist. The following result guarantees that 𝜏a is well defined as an element of the mapping class group. Proposition 2.1. (1) The definition of 𝜏a does not depend on the image of a, or on the mapping a, or on its orientation, or on the choice of aˆ. (2) If a and b are isotopic simple and essential closed curves, then 𝜏a = 𝜏b . Let D = {z ∈ C | |z| ≤ 1} be the standard disc and P2 = { −1 2 , 1 2 }. The standard … view at source ↗
Figure 2.3
Figure 2.3. Standard half-twist. An arc in (Σ, P) is an embedding a : [0, 1] → Σ such that a(0), a(1) ∈ P, a( (0, 1)) ∩ P = ∅, and a( [0, 1]) ∩ 𝜕Σ = ∅. Let a : [0, 1] → Σ be an arc. We choose an embedding aˆ : D → Σ such that aˆ(D) ∩ P = {a(0), a(1)}, aˆ(D) ∩ 𝜕Σ = ∅, and aˆ(t − 1 2 ) = a(t) for each t ∈ [0, 1]. We define Sa ∈ Homeo+ (Σ, P) as follows. Suppose that x ∈ Σ. (1) If x is in the image of aˆ, with x = aˆ(y), then Sa (… view at source ↗
Figures from the paper (16 more)
Figure 2.4
Figure 2.4. Figure 2.4: Half-twist. Note that Sa interchanges the two extremities of a, and, as for Dehn twists, the element 𝜎a is well defined, due to the following result. Proposition 2.2. (1) The definition of 𝜎a only depends on the image of a, and not on the mapping a, or on its orienta…
Figure 2.5
Figure 2.5. Figure 2.5: Regular boundary of an arc. The elements 𝜎a and 𝜏b are connected by the following relation [PITH_FULL_IMAGE:figures/full_fig_p005_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: Coxeter graph An−1 . The example of a mapping class group that interests us in this paper is the one where Σ = D = {z ∈ C | |z| ≤ 1} is the standard disc and Pn = {p1, . . . , pn} is a collection of n points in the interior of D positioned as in [PITH_FULL_IMAGE:fig…
Figure 2.7
Figure 2.7. Figure 2.7: Marked disc. Theorem 2.4. (Artin [Art47]) Let n ≥ 1 and t1, . . . , tn−1 be the standard generators of the braid group Bn . We have an isomorphism Bn → M (D, Pn) that maps ti to 𝜎ei for each 1 ≤ i ≤ n − 1. Example 2.5. Applying Theorem 2.4 with n = 1 we deduce that M…
Figure 2.8
Figure 2.8. Figure 2.8: Mapping class groups of the cylinder with [PITH_FULL_IMAGE:figures/full_fig_p007_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: ). The following result yields a very useful criterion for deciding whether two simple and essential closed curves are in minimal position. D a b [PITH_FULL_IMAGE:figures/full_fig_p007_2_9.png]
Figure 2.10
Figure 2.10. Figure 2.10: We observe that a1 and b do not co-bound any disc, hence, by Theorem 2.9, i( [a1], [b]) = 2. Further, i( [a2], [b]) = 2. Suppose that k1, k2 ∈ Z and g = 𝜏 k1 a1 𝜏 k2 a2 . Applying Corollary 2.11 we have i(g( [b]), [b]) = 4(|k1| + |k2|). Hence, by Proposition 2.8, 𝜏g…
Figure 2.11
Figure 2.11. Figure 2.11: Centraliser of {t1, t2, t4, t5, t6} in B7 . Lemma 2.15. The centraliser of {t1, t2, t4, t5, t6} in B7 is a free abelian group of rank 3 generated by 𝜏a1 , 𝜏a2 and 𝜏d . Remark. Recall that the centre of B7 is the infinite cyclic group generated by Δ 2 = (t1t2 · · · t…
Figure 2.12
Figure 2.12. Figure 2.12: Sub-surface. Proof of Lemma 2.15. It is clear that 𝜏a1 , 𝜏a2 and 𝜏d commute with ti = 𝜎ei for each i ∈ {1, 2, 4, 5, 6}. Further, the fact that 𝜏a1 , 𝜏a2 and 𝜏d generate a free abelian group of rank 3 is a classical result (see [FM12, Lemma 3.17]). It remains to prov…
Figure 2.13
Figure 2.13. Figure 2.13: The curves b1, b2, b4, b5, b6 . Suppose that i ∈ {1, 2, 4, 5, 6}. If G reverses the orientation of bi , then G permutes the two connected components of D \ bi . But this is not possible because one of the two connected components contains 𝜕D, which is fixed pointwis…
Figure 2.14
Figure 2.14. Figure 2.14: Decomposition of D into sub-surfaces. together with results of Castel [Cas16] and Chen–Kordek–Margalit [CKM19] on endomorphisms of braid groups. We begin by recalling these two results. Assume that n ≥ 4 and 2 ≤ p ≤ n − 2. Let Ωn,p be the Coxeter graph shown in [PI…
Figure 3.1
Figure 3.1. Figure 3.1: Coxeter graph Ωn,p . Theorem 3.1. (Baumeister–Neaime–Rees [BNR23]) Let n ≥ 4, and let w be a proper quasi-Coxeter element of W[Dn]. Then there exists p ∈ {2, . . . , n − 2} such that G( [1, w]) is isomorphic to Qn,p . A homomorphism 𝜑 : G → H is cyclic if its image i…
Figure 3.2
Figure 3.2. Figure 3.2: (b). (3) If p = n − 2, then g ∈ ⟨𝜏a1 , 𝜎en−1 , 𝜏d⟩, where a1 and d are the simple closed curves shown in [PITH_FULL_IMAGE:figures/full_fig_p013_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: The curves a1 , a2 and bp . by Proposition 2.8 (1), gt2 pg −1 = 𝜏g( [bp ] ) , and by Proposition 2.8 (3), 𝜏[bp ] and 𝜏g( [bp ] ) commute if and only if i( [bp], g( [bp])) = 0. We conclude that k1 = k2 = 0, hence that g = 1 and 𝜑(s ′ p ) = t 𝜀 p . Case 2: p = 2. By Cl…
Figure 3.4
Figure 3.4. Figure 3.4: The curves b1 , b2 , a2 and c [PITH_FULL_IMAGE:figures/full_fig_p014_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: The curves bp−1 and c. References [All02] D. Allcock. Braid pictures for Artin groups. Trans. Amer. Math. Soc. 354 (2002), no. 9, 3455–3474 [PITH_FULL_IMAGE:figures/full_fig_p015_3_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 1 linked inside Pith

  1. [1]

    D. Allcock. Braid pictures for Artin groups. Trans. Amer. Math. Soc. 354 (2002), no. 9, 3455--3474

    2002

  2. [2]

    E. Artin. Theory of braids. Ann. of Math. (2) 48 (1947), 101--126

    1947

  3. [3]

    Baumeister, M

    B. Baumeister, M. Dyer, C. Stump, P. Wegener. A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements. Proc. Amer. Math. Soc. Ser. B 1 (2014), 149--154

    2014

  4. [4]

    Baumeister, T

    B. Baumeister, T. Gobet, K. Roberts, P. Wegener. On the Hurwitz action in finite Coxeter groups. J. Group Theory 20 (2017), no. 1, 103--131

    2017

  5. [5]

    Baumeister, D.\,F

    B. Baumeister, D.\,F. Holt, G. Neaime, S. Rees. Interval groups related to finite Coxeter groups. Part II. Trans. London Math. Soc. 10 (2023), no. 1, 100--123

    2023

  6. [6]

    Baumeister, D.\,F

    B. Baumeister, D.\,F. Holt, G. Neaime, S. Rees. Isomorphism and non-isomorphism for interval groups of type D_n . J. Algebra 629 (2023), 399--423

    2023

  7. [7]

    Baumeister, G

    B. Baumeister, G. Neaime, S. Rees. Interval groups related to finite Coxeter groups I. Algebr. Comb. 6 (2023), no. 2, 471--506

    2023

  8. [8]

    D. Bessis. The dual braid monoid. Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 647--683

    2003

Show all 25 references
  1. [9]

    D. Bessis. A dual braid monoid for the free group. J. Algebra 302 (2006), no. 1, 55--69

    2006

  2. [10]

    Brady, C

    T. Brady, C. Watt. Non-crossing partition lattices in finite real reflection groups. Trans. Amer. Math. Soc. 360 (2008), no. 4, 1983--2005

    2008

  3. [11]

    F. Castel. Geometric representations of the braid groups. Astérisque (2016), no. 378

    2016

  4. [12]

    R.\,W. Carter. Conjugacy classes in the Weyl group. Compositio Math. 25 (1972), 1--59

    1972

  5. [13]

    L. Chen, K. Kordek, D. Margalit. Homomorphisms between braid groups. Preprint, 2019. arXiv:1910.00712

    2019 arXiv

  6. [14]

    Crisp, L

    J. Crisp, L. Paris. Artin groups of type B and D . Adv. Geom. 5 (2005), no. 4, 607--636

    2005

  7. [15]

    Delucchi, G

    E. Delucchi, G. Paolini, M. Salvetti. Dual structures on Coxeter and Artin groups of rank three. Geom. Topol. 28 (2024), no. 9, 4295--4336

    2024

  8. [16]

    F. Digne. Présentations duales des groupes de tresses de type affine A . Comment. Math. Helv. 81 (2006), no. 1, 23--47

    2006

  9. [17]

    F. Digne. A Garside presentation for Artin-Tits groups of type C_n . Ann. Inst. Fourier (Grenoble) 62 (2012), no. 2, 641--666

    2012

  10. [18]

    D.\,B.\,A. Epstein. Curves on 2-manifolds and isotopies. Acta Math. 115 (1966), 83--107

    1966

  11. [19]

    B. Farb, D. Margalit. A primer on mapping class groups. Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012

    2012

  12. [20]

    Fathi, F

    A. Fathi, F. Laudenbach, V. Poenaru. Travaux de Thurston sur les surfaces -- Séminaire Orsay. Astérisque, no. 66--67 (1979)

    1979

  13. [21]

    McCammond, R

    J. McCammond, R. Sulway. Artin groups of Euclidean type. Invent. Math. 210 (2017), no. 1, 231--282

    2017

  14. [22]

    G. Paolini. The dual approach to the K ( , 1) conjecture. Geometric methods in group theory -- papers dedicated to Ruth Charney, 177--201. Sémin. Congr., 34. Société Mathématique de France, Paris, 2025

    2025

  15. [23]

    Paolini, M

    G. Paolini, M. Salvetti. Proof of the K ( ,1) conjecture for affine Artin groups. Invent. Math. 224 (2021), no. 2, 487--572

    2021

  16. [24]

    Paris, D

    L. Paris, D. Rolfsen. Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521 (2000), 47--83

    2000

  17. [25]

    Perron, J.\,P

    B. Perron, J.\,P. Vannier. Groupe de monodromie géométrique des singularités simples. Math. Ann. 306 (1996), no. 2, 231--245

    1996

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.