Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Endomorphisms of Artin groups of type D

Fabrice Castel, Luis Paris

TL;DR

The paper achieves a complete classification of endomorphisms for the Artin group $A[D_n]$ with $n\ge6$, and determines its automorphism and outer automorphism groups. It extends the analysis to cross-type homomorphisms between $A[D_n]$ and $A[A_{n-1}]$, and to endomorphisms of the quotient $A[D_n]/Z(A[D_n])$, establishing lifting properties and co-Hopfianity. The approach marries algebraic decompositions $A[D_n]\cong \ker(\pi)\rtimes A[A_{n-1}]$ with geometric representations into mapping class groups, leveraging Perron–Vannier-type constructions and curve complex techniques. Together, these results deepen understanding of endomorphisms beyond braid groups, highlighting a robust geometric toolkit for spherical-type Artin groups.

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).

Endomorphisms of Artin groups of type D

TL;DR

The paper achieves a complete classification of endomorphisms for the Artin group with , and determines its automorphism and outer automorphism groups. It extends the analysis to cross-type homomorphisms between and , and to endomorphisms of the quotient , establishing lifting properties and co-Hopfianity. The approach marries algebraic decompositions with geometric representations into mapping class groups, leveraging Perron–Vannier-type constructions and curve complex techniques. Together, these results deepen understanding of endomorphisms beyond braid groups, highlighting a robust geometric toolkit for spherical-type Artin groups.

Abstract

In this paper we determine a classification of the endomorphisms of the Artin group of type for . In particular we determine its automorphism group and its outer automorphism group. We also determine a classification of the homomorphisms from to the Artin group of type and a classification of the homomorphisms from to for . We show that any endomorphism of the quotient lifts to an endomorphism of for . We deduce a classification of the endomorphisms of , we determine the automorphism and outer automorphism groups of , and we show that is co-Hopfian, for . The results are algebraic in nature but the proofs are based on topological arguments (curves on surfaces and mapping class groups).
Paper Structure (7 sections, 28 theorems, 46 equations, 21 figures)

This paper contains 7 sections, 28 theorems, 46 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Definitions and statements
  3. Geometric representations
  4. Homomorphisms from $A[D_n]$ to $A[A_{n-1}]$
  5. Homomorphisms from $A[A_{n-1}]$ to $A[D_n]$
  6. Endomorphisms of $A[D_n]$
  7. Endomorphisms of $A [D_n] / Z (A [D_n])$

Key Result

Theorem 2.1

Let $n\ge 5$. Let $\varphi:A[D_n]\to A[A_{n-1}]$ be a homomorphism. Then up to conjugation we have one of the following two possibilities.

Figures (21)

  • Figure 1. 1: Coxeter graph $A_n$
  • Figure 1. 2: Coxeter graph $D_n$
  • Figure 3. 1: Geometric representation of $A[A_{n-1}]$
  • Figure 3. 2: Geometric representation of $A[D_n]$
  • Figure 3. 3: The loop $f_{n-1}\in\pi_1(\Sigma_n,x)$
  • ...and 16 more figures

Theorems & Definitions (49)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • Proposition 2.7
  • Theorem 2.8
  • ...and 39 more