Endomorphisms of Artin groups of type D

TL;DR

This paper classifies endomorphisms and automorphisms of Artin groups of type D, explores their relationships with type A groups, and proves structural properties using algebraic and topological methods.

Contribution

It provides a comprehensive classification of endomorphisms and automorphisms of Artin groups of type D, including their quotients, and establishes new structural properties.

Findings

Automorphism group of A[D_n] determined

Endomorphisms of A[D_n]/Z(A[D_n]) classified

A[D_n]/Z(A[D_n]) shown to be co-Hopfian for n≥6

Abstract

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).