Endomorphisms of Artin groups of type D

Fabrice Castel; Luis Paris

doi:10.2140/agt.2025.25.3975

paper

Endomorphisms of Artin groups of type D

arXiv:2307.02880 · doi:10.2140/agt.2025.25.3975

Abstract

In this paper 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).