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arxiv: 2406.02484 · v4 · submitted 2024-06-04 · 🧮 math.GR

Endomorphisms of Artin groups of type tilde A_n

Luis Paris , Ignat Soroko This is my paper

Pith reviewed 2026-05-24 00:13 UTC · model grok-4.3

classification 🧮 math.GR
keywords Artin groupsendomorphismsaffine typegroup classificationbraid relationshomomorphisms
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The pith

Artin groups of affine type Ã_n for n at least 4 have all their endomorphisms classified.

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

The paper establishes a complete list of all endomorphisms for these groups when n is four or larger. Endomorphisms are the homomorphisms from the group to itself, so the classification describes every possible way to map the group into itself while preserving its relations. A reader would care because these groups generalize the braid group, and knowing their self-maps clarifies their structure and possible actions. The result applies specifically to the affine case, which has a cycle in its defining diagram.

Core claim

The authors determine a classification of the endomorphisms of the Artin group of affine type Ã_n for n≥4 by examining how any such map must send the standard generators while respecting the braid relations in the group's presentation.

What carries the argument

The standard presentation of the Ã_n Artin group, whose generators and braid-like relations constrain the possible images of each generator under an endomorphism.

If this is right

  • The invertible endomorphisms yield the automorphism group as a special case.
  • All endomorphisms factor in ways compatible with the cyclic diagram of the affine type.
  • The classification separates maps that preserve the center from those that do not.

Where Pith is reading between the lines

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

  • The result may extend to deciding whether these groups are Hopfian or have other finiteness properties.
  • One could test the classification computationally for small n by enumerating maps on generators that satisfy the relations.

Load-bearing premise

The group is defined exactly by its usual presentation, with no extra relations that would further restrict which maps on generators extend to endomorphisms.

What would settle it

An explicit endomorphism of the group for some n≥4 that falls outside every class listed in the classification.

Figures

Figures reproduced from arXiv: 2406.02484 by Ignat Soroko, Luis Paris.

Figure 1.1
Figure 1.1. Figure 1.1: Coxeter graph A˜ n , n ⩾ 2 The aim of the present paper is to determine a classification of the endomorphisms of the Artin groups of type A˜ n for n ⩾ 4, where A˜ n is the affine Coxeter graph depicted in [PITH_FULL_IMAGE:figures/full_fig_p002_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Coxeter graphs An and Bn Let Γ be a Coxeter graph. For X ⊂ S we denote by ΓX the full subgraph of Γ spanned by X, by AX[Γ] the subgroup of A[Γ] generated by X, and by WX[Γ] the subgroup of W[Γ] generated by X. We know by [Lek83] that AX[Γ] is naturally isomorphic to A[ΓX], and we know by [Bou68, Chapter 4, Section 1.8, Theorem 2(i)] that WX[Γ] is naturally isomorphic to W[ΓX]. A subgroup of the form AX[Γ… view at source ↗
Figure 2.2
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. Figure 3.1: A half-twist The following is known to experts but, as far as we know, the proof is not readily available in the literature, so we provide it here. Proposition 3.5 Let Σ be an oriented compact surface and let P be a finite collection of marked points in the interior of Σ. If |P| = 3, or if |P| = 4 and the genus of Σ is zero, we require additionally that ∂Σ ̸= ∅. Let a, b be two non-isotopic arcs of (Σ,P)… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: A disk with marked points reduction class for f . A reduction class [a] is an essential reduction class if, for each [b] ∈ C(Σ,P) such that i([a], [b]) ̸= 0 and for each m ∈ Z \ {0}, we have f m([b]) ̸= [b]. In particular, if [a] is an essential reduction class and [b] is any reduction class, then i([a], [b]) = 0. We denote by S(f) the set of essential reduction classes for f . The following gathers toge… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Arcs in the disk with marked points Proposition 4.1 Let n ⩾ 4. Let φ: A[A˜ n] → A[An+1] be a homomorphism. Then we have one of the following three possibilities. (1) φ is cyclic. (2) There exist g ∈ A[An+1], k ∈ {0, 1}, ε ∈ {±1} and q ∈ Z such that φ(ti) = gs′ε i ∆2qg −1 for all 1 ⩽ i ⩽ n, and φ(t0) = guε k∆2qg −1 . (3) There exist g ∈ A[An+1], k ∈ {0, 1}, ε ∈ {±1} and p, q ∈ Z such that φ(ti) = gs′ε i ∆… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: A circle in the disk with marked points Since t0 is conjugate to t1 in A[A˜ n], φ ′ (t0) is conjugate to φ ′ (t1) in A[An+1]. Hence φ ′ (t0) is of the form φ ′ (t0) = H ε a ′T p d ′T q ∂D , where d ′ is a circle which bounds a disk containing n + 1 marked points, and a ′ is an arc inside this disk. Moreover, if aˆ ′ is the boundary of a regular neighborhood of a ′ , then S(φ ′ (t0)) = {[aˆ ′ ], [d ′ ]}. … view at source ↗
read the original abstract

We determine a classification of the endomorphisms of the Artin group of affine type $\tilde A_n$ for $n\ge 4$.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript determines a classification of the endomorphisms of the Artin group of affine type Ã_n for n≥4, using the standard Coxeter presentation and the approach of mapping generators while preserving the braid relations.

Significance. If the classification is exhaustive and correct, the result contributes to the study of endomorphism monoids of Artin groups, extending existing work on spherical and other affine types. The classification may aid in understanding automorphism groups and related geometric group theory questions.

minor comments (2)
  1. The abstract provides no outline of the proof strategy or key lemmas; adding a brief sketch in the introduction would improve readability.
  2. Notation for the generators and relations of the Ã_n Artin group should be stated explicitly in §1 or §2 for self-contained reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and positive recommendation of minor revision. The report provides no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper claims a classification of endomorphisms of the Artin group of affine type Ã_n (n≥4) via the standard Coxeter presentation and generator mappings that preserve the defining braid relations. No equations, fitted parameters, self-definitional constructions, or load-bearing self-citations appear in the provided abstract or described method. The approach is the conventional one for such groups and does not reduce any central claim to its own inputs by construction; the result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the result rests on standard definitions of Artin groups.

pith-pipeline@v0.9.0 · 5527 in / 1034 out tokens · 21668 ms · 2026-05-24T00:13:53.945004+00:00 · methodology

discussion (0)

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Reference graph

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