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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.02880 |
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Table of Contents:
- 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).