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Endomorphisms of Artin groups of type $B_n$

Luis Paris, Ignat Soroko

TL;DR

This work delivers a complete endomorphism classification for the spherical-type Artin group $A[B_n]$ with $n\ge5$ and for its center quotient, up to conjugacy. The authors leverage a fixed chain of inclusions $A[\tilde{A}_{n-1}]\hookrightarrow A[B_n]\hookrightarrow A[A_n]$ and a prior Paris–Soroko classification of maps $A[\tilde{A}_{n-1}]\to A[A_n]$, augmented by algebraic and mapping-class techniques to express endomorphisms in explicit normal forms involving commuting elements, powers of Garside elements, and Dehn-twist-like constructs ($\Delta_B$, $\Delta_Y$). The main results yield four families of endomorphisms for $A[B_n]$ (the last being a mixed type with $t_i$ mapped via $\Delta_Y^{2p}$ and $\Delta_B^q$), along with a detailed description of the endomorphisms of the quotient $\overline{A[B_n]}$ and the resulting automorphism/outer-automorphism structure, Hopfian and co-Hopfian properties, and the characteristic status of the subgroups. These findings connect algebraic properties of Artin groups with mapping-class-group techniques and provide a framework for understanding endomorphisms in broader spherical and affine Artin types, including explicit embeddings and centralization phenomena that inform potential extensions to automorphism groups.

Abstract

We determine a classification of the endomorphisms of the Artin groups of spherical type $B_n$ for $n\ge 5$, and of their quotients by the center.

Endomorphisms of Artin groups of type $B_n$

TL;DR

This work delivers a complete endomorphism classification for the spherical-type Artin group with and for its center quotient, up to conjugacy. The authors leverage a fixed chain of inclusions and a prior Paris–Soroko classification of maps , augmented by algebraic and mapping-class techniques to express endomorphisms in explicit normal forms involving commuting elements, powers of Garside elements, and Dehn-twist-like constructs (, ). The main results yield four families of endomorphisms for (the last being a mixed type with mapped via and ), along with a detailed description of the endomorphisms of the quotient and the resulting automorphism/outer-automorphism structure, Hopfian and co-Hopfian properties, and the characteristic status of the subgroups. These findings connect algebraic properties of Artin groups with mapping-class-group techniques and provide a framework for understanding endomorphisms in broader spherical and affine Artin types, including explicit embeddings and centralization phenomena that inform potential extensions to automorphism groups.

Abstract

We determine a classification of the endomorphisms of the Artin groups of spherical type for , and of their quotients by the center.
Paper Structure (10 sections, 20 theorems, 74 equations, 6 figures)

This paper contains 10 sections, 20 theorems, 74 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definitions and statements
  3. Key definitions
  4. Results on endomorphisms of A[Bn]
  5. Results on endomorphisms of A[Bn]/Z(A[Bn])
  6. Preliminaries
  7. Proofs for endomorphisms of A[Bn]
  8. Proofs for endomorphisms of A[Bn]/Z(A[Bn])
  9. Additional remarks
  10. Some open questions

Key Result

Theorem 2.1

Let $n\geqslant5$ and $\varphi\colon A[B_{n}]\longrightarrow A[B_{n}]$ be an endomorphism. Then, up to conjugation, $\varphi$ belongs to one of the following three types, written in generators $t_0,\dots,t_{n-1},\rho_B$ as follows:

Figures (6)

  • Figure 1. 1: The Coxeter graph of type $B_n$ ($n\geqslant2$)
  • Figure 2. 1: Coxeter graphs $A_n$ ($n\geqslant1$), and $\tilde{A}_{n-1}$ ($n\geqslant3$)
  • Figure 2. 2: Braid pictures of the standard generators $t_1,\dots,t_{n-1}$ and elements $v_0$ and $v_1$, depicted for $n=4$.
  • Figure 3. 1: A half-twist
  • Figure 3. 2: Disk $\mathbb D$ with punctures $p_i$ (denoted for the ease of notation by $i$), $0\leqslant i\leqslant n$.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Theorem 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Remark 1
  • Corollary 2.4
  • Corollary 2.5: Charney--Crisp ChaCri1, Bell--Margalit BelMar2
  • Corollary 2.6
  • Remark 2
  • Proposition 2.7
  • Theorem 2.8
  • ...and 28 more