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Pré-Publication, Document De Travail Année : 2024

Endomorphisms of Artin groups of type D

Résumé

In this paper we determine a classification of the endomorphisms of the Artin group $A [D_n]$ of type $D_n$ for $n\ge 6$. 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 $n\ge 6$. We show that any endomorphism of the quotient $A [D_n] / Z (A [D_n])$ lifts to an endomorphism of $A [D_n]$ for $n \ge 4$. 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 $n \ge 6$. The results are algebraic in nature but the proofs are based on topological arguments (curves on surfaces and mapping class groups).
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Dates et versions

hal-04152823 , version 1 (05-07-2023)
hal-04152823 , version 2 (28-08-2023)
hal-04152823 , version 3 (04-03-2024)
hal-04152823 , version 4 (21-05-2024)

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Citer

Fabrice Castel, Luis Paris. Endomorphisms of Artin groups of type D. 2024. ⟨hal-04152823v4⟩
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