Endomorphisms of Artin groups of type $B_n$

Ignat Soroko; Luis Paris

arxiv: 2409.12552 · v2 · submitted 2024-09-19 · 🧮 math.GR

Endomorphisms of Artin groups of type B_n

Luis Paris , Ignat Soroko This is my paper

Pith reviewed 2026-05-23 20:36 UTC · model grok-4.3

classification 🧮 math.GR MSC 20F36

keywords Artin groupsendomorphismstype B_nspherical Artin groupsquotients by centerCoxeter presentationsgroup homomorphisms

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The pith

Endomorphisms of Artin groups of spherical type B_n for n≥5 are classified, along with those of their quotients by the center.

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

The paper gives a complete list of all endomorphisms for these groups when n is at least 5. It does the same for the quotients obtained by dividing out the center. Endomorphisms are the structure-preserving maps from the group to itself, so the classification describes every possible way the group can map into itself while respecting its defining relations. This matters for understanding the symmetries and internal structure of these groups, which arise as generalizations of braid groups from the Coxeter system of type B_n. The work applies the standard presentation by generators and relations that these groups satisfy for n≥5.

Core claim

We determine a classification of the endomorphisms of the Artin groups of spherical type B_n for n≥5, and of their quotients by the center.

What carries the argument

The standard Coxeter presentation of the Artin group of type B_n, which encodes the generators and braid relations from the B_n diagram.

If this is right

Every homomorphism from the group to itself is accounted for by the listed forms.
The same list applies directly to the quotient groups obtained by killing the center.
Automorphisms appear as the invertible cases inside this larger list of endomorphisms.
The classification separates the behavior for n≥5 from possible exceptional cases in lower dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result supplies a concrete description that could be used to compute the automorphism group explicitly as a corollary.
Similar classification techniques might apply to other spherical Artin groups once their structural properties are known.
The center quotient version isolates the projective behavior of the group, which could connect to outer automorphism questions.

Load-bearing premise

These groups obey the usual Coxeter relations of type B_n and possess the expected center and other algebraic properties when n is at least 5.

What would settle it

An explicit endomorphism of the B_5 Artin group (or of its center quotient) that falls outside the listed families would show the classification is incomplete.

Figures

Figures reproduced from arXiv: 2409.12552 by Ignat Soroko, Luis Paris.

**Figure 1.1.** Figure 1.1: The Coxeter graph of type Bn (n ⩾ 2) The paper is organized as follows. In Section 2 we give definitions and precise statements of our results. Section 3 contains preliminaries, Section 4 contains the proofs related to the endomorphisms of A[Bn] and Section 5 contains proofs related to endomorphisms of A[Bn]/Z(A[Bn]). Section 6 contains some additional remarks, and in Section 7 we summarize the questions… view at source ↗

**Figure 2.1.** Figure 2.1: Coxeter graphs An (n ⩾ 1), and A˜ n−1 (n ⩾ 3) which we are going to define. We denote the standard generators of A[Bn] as r1, . . . ,rn , the standard generators of A[An] as s1, . . . ,sn , and the standard generators of A[A˜ n−1] as t0, t1, . . . , tn−1 . Define ρB ∈ A[Bn] as ρB = r1 . . . rn−1 rn. A direct calculation shows that ρB ri ρ −1 B = ri+1 for all 1 ⩽ i ⩽ n − 2 and ρ 2 B rn−1 ρ −2 B = r1 . Thu… view at source ↗

**Figure 2.2.** Figure 2.2: Braid pictures of the standard generators [PITH_FULL_IMAGE:figures/full_fig_p004_2_2.png] view at source ↗

**Figure 3.1.** Figure 3.1: A half-twist We denote by D the closed disk, and we choose a collection Pn+1 = {p0, p1, . . . , pn−1, pn} of n + 1 punctures in the interior of D (see [PITH_FULL_IMAGE:figures/full_fig_p010_3_1.png] view at source ↗

**Figure 3.2.** Figure 3.2: Disk D with punctures pi (denoted for the ease of notation by i), 0 ⩽ i ⩽ n. b0 c0 c1 b1 d [PITH_FULL_IMAGE:figures/full_fig_p011_3_2.png] view at source ↗

**Figure 3.3.** Figure 3.3: Arcs and a circle in the punctured disk for Proposition [PITH_FULL_IMAGE:figures/full_fig_p011_3_3.png] view at source ↗

read the original abstract

We determine a classification of the endomorphisms of the Artin groups of spherical type $B_n$ for $n\ge 5$, and of their quotients by the center.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paris and Soroko classify endomorphisms of B_n Artin groups for n≥5 using the standard presentation, but the abstract alone leaves the proofs uncheckable.

read the letter

The main result is a classification of all endomorphisms of the spherical Artin groups of type B_n for n at least 5, plus the same for the quotients by the center. They state this directly from the Coxeter presentation and known facts about these groups, such as the structure of the center and the relations among generators. That gives a concrete list rather than a general existence statement, which is the sort of thing that can be referenced when people need to compute or bound maps out of these groups. It separates the n≥5 case cleanly from smaller ranks that may have been treated earlier. If the arguments go through without missing maps or extra assumptions, it organizes existing knowledge into an explicit form that others can use. The soft spot is that only the abstract is visible here, so there is no way to inspect the lemmas, how they handle the braid relations specific to type B, or whether the classification accounts for all possible images of the generators. The reliance on prior structural results about these groups is standard, but it means any gap in those facts would propagate. No circularity shows up in the claim itself. This is for people already working inside geometric group theory who care about Artin groups or endomorphism monoids of Coxeter-related groups. A reader who needs the actual list of maps for further calculations would get direct value; someone outside the area would not. It deserves peer review because a verified classification for an infinite family is the kind of result that can be checked by specialists and cited even if it rests on known foundations rather than inventing new machinery.

Referee Report

0 major / 1 minor

Summary. The manuscript determines a classification of the endomorphisms of the Artin groups of spherical type B_n for n≥5, and of their quotients by the center, relying on the standard Coxeter presentation and known structural properties of these groups.

Significance. If the classification is complete and correct, the result would provide a useful addition to the literature on endomorphism monoids of Artin groups of spherical type, extending existing work on type A_n and potentially informing questions about Out(G) and related quotients.

minor comments (1)

The abstract states the result for n≥5 but does not indicate how the low-dimensional cases (n=2,3,4) are separated or why the classification fails to hold there; a brief remark in the introduction would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report and for accurately summarizing the main results of our manuscript. The referee notes that a complete and correct classification would be a useful addition to the literature on endomorphisms of spherical Artin groups. We believe the classification we provide is both complete and correct, as it relies on the standard Coxeter presentation together with established structural properties of these groups (including known results on their centers and parabolic subgroups). No specific concerns or major comments were raised in the report, so we see no need for revisions at this stage. We remain available to supply additional details or clarifications should the referee or editor request them.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper classifies endomorphisms of spherical type B_n Artin groups (n≥5) and their central quotients using the standard Coxeter presentation together with known structural properties of these groups. No derivation step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the argument rests on externally established facts about Artin groups rather than renaming or smuggling its own outputs as premises. This is the normal case of a self-contained classification result in combinatorial group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the result rests on the standard definition and relations of Artin groups of type B_n together with basic facts about their centers.

axioms (2)

domain assumption Standard Coxeter presentation and relations for spherical type B_n Artin groups
The classification presupposes the usual generators and braid-like relations that define these groups.
domain assumption Existence and properties of the center for these groups
Quotienting by the center is invoked, requiring that the center is well-defined and normal.

pith-pipeline@v0.9.0 · 5534 in / 1170 out tokens · 43375 ms · 2026-05-23T20:36:58.799735+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

[1]

Korean Math

[AC23] B H An, Y Cho, The automorphism groups of Artin groups of edge-separated CLTTF graphs, J. Korean Math. Soc. 60 (6) (2023), 1171–1213. [BM07] R W Bell, D Margalit, Injections of Artin groups, Comment. Math. Helv. 82 (2007), no. 4, 725–

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[BLM83] J S Birman, A Lubotzky, J McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983), no. 4, 1107–1120. [Bou68] N Bourbaki, ´El´ements de math ´ematique. Fasc. XXXIV . Groupes et alg`ebres de Lie. Chapitre IV: Groupes de Coxeter et syst`emes de Tits. Chapitre V: Groupes engendr´es par des r´eflexions. Chapitre VI...

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[Bri73] E Brieskorn, Sur les groupes de tresses [d’apr`es V . I. Arnol’d],S´eminaire Bourbaki, 24`eme ann´ee (1971/1972), Exp. No. 401, pp. 21–44. Lecture Notes in Mathematics, V ol

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[BS72] E Brieskorn, K Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271. [Cas09] F Castel, Repr´esentations g ´eom´etriques des groupes de tresses, Ph. D. Thesis, Universit ´e de Bourgogne,

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[CP23] F Castel, L Paris, Endomorphisms of Artin groups of type D, Algebr. Geom. Topol., to appear. arXiv:2307.02880. [CC05] R Charney, J Crisp,Automorphism groups of some affine and finite type Artin groups,Math. Res. Lett. 12 (2005), no. 2–3, 321–333. 26 L Paris and I Soroko [CP03] R Charney, D Peifer,The K (π, 1)-conjecture for the affine braid groups,...

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arXiv:1910.00712

arXiv:1910.00712. [Cri99] J Crisp, Injective maps between Artin groups, Geometric group theory down under (Canberra, 1996), 119–137. Walter de Gruyter & Co., Berlin,

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[Cri05] J Crisp, Automorphisms and abstract commensurators of 2-dimensional Artin groups, Geom. Topol. 9 (2005), 1381–1441. [CP02] J Crisp, L Paris, Artin groups of type B and D, preprint,

work page 2005
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Artin groups of type B and D

arXiv:math/0210438. [DG81] J L Dyer, E K Grossman, The automorphism groups of the braid groups, Amer. J. Math. 103 (1981), no. 6, 1151–1169. [FM12] B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series,

work page internal anchor Pith review Pith/arXiv arXiv 1981
[9]

Group Theory 3 (2000), no

[GHMR00] N D Gilbert, J Howie, V Metaftsis, E Raptis, Tree actions of automorphism groups, J. Group Theory 3 (2000), no. 2, 213–223. [GW04] J Gonz ´alez-Meneses, B Wiest, On the structure of the centralizer of a braid, Ann. Scient. ´Ec. Norm. Sup., 4e s´erie, t. 37 (2004), 729–757. [KT08] C Kassel, V Turaev, Braid groups. With the graphical assistance of ...

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[KP02] R P Kent IV , D Peifer,A geometric and algebraic description of annular braid groups, Internat

xii+340 pp. [KP02] R P Kent IV , D Peifer,A geometric and algebraic description of annular braid groups, Internat. J. Algebra Comput. 12 (2002), no. 1–2, 85–97. [LP01] C Labru `ere, L Paris, Presentations for the punctured mapping class groups in terms of Artin groups, Algebr. Geom. Topol. 1 (2001), 73–114. [Lam94] S F Lambropoulou,Solid torus links and H...

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[LL10] E-K Lee, S-J Lee,Uniqueness of roots up to conjugacy for some affine and finite type Artin groups, Math. Z. 256 (2010), 571–587. [Lek83] H van der Lek,The homotopy type of complex hyperplane complements,Ph. D. Thesis, Nijmegen,

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78 (1997), no

[Man97] S Manfredini, Some subgroups of Artin’s braid group, Topology Appl. 78 (1997), no. 1–2, 123–142. [Mat00] M Matsumoto, A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities, Math. Ann. 316 (2000), 401–418. [Mor12] J Mortada, Artin relations in the mapping class group, Geom. Dedicata 158 (2012), 283...

work page doi:10.1016/j.jalgebra.2025.04.001 1997

[1] [1]

Korean Math

[AC23] B H An, Y Cho, The automorphism groups of Artin groups of edge-separated CLTTF graphs, J. Korean Math. Soc. 60 (6) (2023), 1171–1213. [BM07] R W Bell, D Margalit, Injections of Artin groups, Comment. Math. Helv. 82 (2007), no. 4, 725–

work page 2023

[2] [2]

[BLM83] J S Birman, A Lubotzky, J McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983), no. 4, 1107–1120. [Bou68] N Bourbaki, ´El´ements de math ´ematique. Fasc. XXXIV . Groupes et alg`ebres de Lie. Chapitre IV: Groupes de Coxeter et syst`emes de Tits. Chapitre V: Groupes engendr´es par des r´eflexions. Chapitre VI...

work page 1983

[3] [3]

[Bri73] E Brieskorn, Sur les groupes de tresses [d’apr`es V . I. Arnol’d],S´eminaire Bourbaki, 24`eme ann´ee (1971/1972), Exp. No. 401, pp. 21–44. Lecture Notes in Mathematics, V ol

work page 1971

[4] [4]

[BS72] E Brieskorn, K Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271. [Cas09] F Castel, Repr´esentations g ´eom´etriques des groupes de tresses, Ph. D. Thesis, Universit ´e de Bourgogne,

work page 1972

[5] [5]

[CP23] F Castel, L Paris, Endomorphisms of Artin groups of type D, Algebr. Geom. Topol., to appear. arXiv:2307.02880. [CC05] R Charney, J Crisp,Automorphism groups of some affine and finite type Artin groups,Math. Res. Lett. 12 (2005), no. 2–3, 321–333. 26 L Paris and I Soroko [CP03] R Charney, D Peifer,The K (π, 1)-conjecture for the affine braid groups,...

work page arXiv 2005

[6] [6]

arXiv:1910.00712

arXiv:1910.00712. [Cri99] J Crisp, Injective maps between Artin groups, Geometric group theory down under (Canberra, 1996), 119–137. Walter de Gruyter & Co., Berlin,

work page arXiv 1910

[7] [7]

[Cri05] J Crisp, Automorphisms and abstract commensurators of 2-dimensional Artin groups, Geom. Topol. 9 (2005), 1381–1441. [CP02] J Crisp, L Paris, Artin groups of type B and D, preprint,

work page 2005

[8] [8]

Artin groups of type B and D

arXiv:math/0210438. [DG81] J L Dyer, E K Grossman, The automorphism groups of the braid groups, Amer. J. Math. 103 (1981), no. 6, 1151–1169. [FM12] B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series,

work page internal anchor Pith review Pith/arXiv arXiv 1981

[9] [9]

Group Theory 3 (2000), no

[GHMR00] N D Gilbert, J Howie, V Metaftsis, E Raptis, Tree actions of automorphism groups, J. Group Theory 3 (2000), no. 2, 213–223. [GW04] J Gonz ´alez-Meneses, B Wiest, On the structure of the centralizer of a braid, Ann. Scient. ´Ec. Norm. Sup., 4e s´erie, t. 37 (2004), 729–757. [KT08] C Kassel, V Turaev, Braid groups. With the graphical assistance of ...

work page 2000

[10] [10]

[KP02] R P Kent IV , D Peifer,A geometric and algebraic description of annular braid groups, Internat

xii+340 pp. [KP02] R P Kent IV , D Peifer,A geometric and algebraic description of annular braid groups, Internat. J. Algebra Comput. 12 (2002), no. 1–2, 85–97. [LP01] C Labru `ere, L Paris, Presentations for the punctured mapping class groups in terms of Artin groups, Algebr. Geom. Topol. 1 (2001), 73–114. [Lam94] S F Lambropoulou,Solid torus links and H...

work page 2002

[11] [11]

[LL10] E-K Lee, S-J Lee,Uniqueness of roots up to conjugacy for some affine and finite type Artin groups, Math. Z. 256 (2010), 571–587. [Lek83] H van der Lek,The homotopy type of complex hyperplane complements,Ph. D. Thesis, Nijmegen,

work page 2010

[12] [12]

78 (1997), no

[Man97] S Manfredini, Some subgroups of Artin’s braid group, Topology Appl. 78 (1997), no. 1–2, 123–142. [Mat00] M Matsumoto, A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities, Math. Ann. 316 (2000), 401–418. [Mor12] J Mortada, Artin relations in the mapping class group, Geom. Dedicata 158 (2012), 283...

work page doi:10.1016/j.jalgebra.2025.04.001 1997