Endomorphisms of Artin groups of type D

Castel, Fabrice; Paris, Luis

doi:10.2140/agt.2025.25.3975

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Endomorphisms of Artin groups of type $D$

Fabrice Castel and Luis Paris

Algebraic & Geometric Topology 25 (2025) 3975–4008

DOI: 10.2140/agt.2025.25.3975

Abstract

We determine a classification of the endomorphisms of the Artin group $A [D_{n}]$ of type $D_{n}$ for $n \geq 6$ . In particular we determine its automorphism group and its outer automorphism group. We also determine a classification of the homomorphisms from $A [D_{n}]$ to the Artin group $A [A_{n - 1}]$ of type $A_{n - 1}$ and a classification of the homomorphisms from $A [A_{n - 1}]$ to $A [D_{n}]$ for $n \geq 6$ . We show that any endomorphism of the quotient $A [D_{n}] ∕ Z (A [D_{n}])$ lifts to an endomorphism of $A [D_{n}]$ for $n \geq 4$ . We deduce a classification of the endomorphisms of $A [D_{n}] ∕ Z (A [D_{n}])$ , we determine the automorphism and outer automorphism groups of $A [D_{n}] ∕ Z (A [D_{n}])$ , and we show that $A [D_{n}] ∕ Z (A [D_{n}])$ is co-Hopfian for $n \geq 6$ . The results are algebraic in nature but the proofs are based on topological arguments (curves on surfaces and mapping class groups).

Keywords

Artin groups of type $D$, endomorphisms, automorphisms, mapping class groups

Mathematical Subject Classification

Primary: 20F36

Secondary: 57K20

References

Publication

Received: 24 August 2023

Revised: 21 May 2024

Accepted: 19 August 2024

Published: 29 October 2025

Authors

Fabrice Castel
IMB, UMR 5584, CNRS Université Bourgogne Europe Dijon France

Luis Paris
Institut de Mathématiques de Bourgogne Université de Bourgogne UMR 5584 du CNRS BP 47870 21078 Dijon France

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Volume 25, issue 7 (2025)