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Homomorphisms between XL-type Artin groups

Martín Blufstein, Alexandre Martin, Nicolas Vaskou

TL;DR

The paper addresses endomorphisms and automorphisms of XL-type Artin groups, establishing that generic XL-type groups are both hopfian and co-hopfian and providing a sharp description of homomorphisms in the free-of-infinity (complete graph) case. It leverages geometric methods on the Deligne complex and the Cycle of Standard Trees Property to constrain images of standard generators, obtaining a complete structure theorem for complete graphs and a co-hopfian criterion for XXXL-type graphs (labels ≥6). It further shows finite generation of Automorphism groups under mild graph conditions and characterizes when Outer automorphism groups are finite, with explicit generating sets involving conjugations, graph automorphisms, global inversion, and edge-twists. These results have implications for rigidity phenomena and the Isomorphism Problem in large-type Artin groups, highlighting how graph structure governs endomorphisms and automorphisms. Overall, the work provides a unified geometric framework for understanding how presentation graphs dictate endomorphism behavior and group automorphisms in XL- and XXXL-type Artin groups.

Abstract

We study homomorphisms between XL-type Artin groups and show that, in a suitable sense, a generic Artin group is both hopfian and co-hopfian. For XL-type Artin groups over complete graphs, we describe all possible homomorphisms with sufficiently large image, and prove in particular that such groups are both hopfian and co-hopfian. For Artin groups over general graphs with all labels at least $6$, we characterise in terms of the presentation graph exactly when these groups are co-hopfian, as well as when they have a finite outer automorphism group. When in addition the presentation graph has no cut-vertex, we show that their automorphism group is finitely generated and we provide a generating set.

Homomorphisms between XL-type Artin groups

TL;DR

The paper addresses endomorphisms and automorphisms of XL-type Artin groups, establishing that generic XL-type groups are both hopfian and co-hopfian and providing a sharp description of homomorphisms in the free-of-infinity (complete graph) case. It leverages geometric methods on the Deligne complex and the Cycle of Standard Trees Property to constrain images of standard generators, obtaining a complete structure theorem for complete graphs and a co-hopfian criterion for XXXL-type graphs (labels ≥6). It further shows finite generation of Automorphism groups under mild graph conditions and characterizes when Outer automorphism groups are finite, with explicit generating sets involving conjugations, graph automorphisms, global inversion, and edge-twists. These results have implications for rigidity phenomena and the Isomorphism Problem in large-type Artin groups, highlighting how graph structure governs endomorphisms and automorphisms. Overall, the work provides a unified geometric framework for understanding how presentation graphs dictate endomorphism behavior and group automorphisms in XL- and XXXL-type Artin groups.

Abstract

We study homomorphisms between XL-type Artin groups and show that, in a suitable sense, a generic Artin group is both hopfian and co-hopfian. For XL-type Artin groups over complete graphs, we describe all possible homomorphisms with sufficiently large image, and prove in particular that such groups are both hopfian and co-hopfian. For Artin groups over general graphs with all labels at least , we characterise in terms of the presentation graph exactly when these groups are co-hopfian, as well as when they have a finite outer automorphism group. When in addition the presentation graph has no cut-vertex, we show that their automorphism group is finitely generated and we provide a generating set.

Paper Structure

This paper contains 23 sections, 65 theorems, 56 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Statement of results.
  4. Free-of-infinity Artin groups.
  5. XXXL-type Artin groups.
  6. Strategy of the proofs and structure of the paper.
  7. Generalisations and necessity of the hypotheses.
  8. Acknowledgements.
  9. Background on Artin groups
  10. Artin groups and their Deligne complexes
  11. Standard trees and the coned-off Deligne complex
  12. Centralisers of elements
  13. Dihedral Artin groups
  14. The image of a standard generator
  15. Configurations of standard trees
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $f: A_\Gamma \rightarrow A_{\Gamma'}$ be a homomorphism between XL-type free-of-infinity Artin groups with rank at least $3$, such that the image of $f$ is not cyclic or contained in a subgroup of $A_{\Gamma'}$ that virtually splits non-trivially as a direct product. Let $\Gamma_1 \subseteq \Gam In particular, the image of $f$ is the parabolic subgroup $gA_{\Gamma_1'}g^{-1}$ of $A_{\Gamma'}$.

Figures (8)

  • Figure 1: The triangle formed by the standard trees $\mathrm{Fix}(s)$, $\mathrm{Fix}(t)$ and $\mathrm{Fix}(trt^{-1})$ is not contained in a translate of the fundamental domain.
  • Figure 2: The graph $X_n$ and its dual tree $T_n$ when $n = 3$.
  • Figure 3: An illustration of the arguments used in the proof of Lemma \ref{['lem:decomposition_in_induced_cycles']}.
  • Figure 4: Going from a triangular structure to a polygonal structure by erasing type $0$ vertices. Type $2$ vertices are drawn in black and type $1$ vertices in white.
  • Figure 5: The inner and outer paths of a corner-cell $P$.
  • ...and 3 more figures

Theorems & Definitions (137)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Example 1.9
  • Example 1.10
  • Example 1.11
  • ...and 127 more