[Paper Review] Artin groups of type B and D

John Crisp, L. Paris|ArXiv.org|Oct 29, 2002

Geometric and Algebraic Topology11 references11 citations

TL;DR

This paper establishes that Artin groups of type $B_n$ and $D_n$ admit semidirect product structures $F \rtimes \mathcal{B}_n$, where $F$ is a free group and $\mathcal{B}_n$ is the $n$-string braid group. The action of $\mathcal{B}_n$ on $F$ arises naturally from fibrations of regular orbit spaces, and the outer automorphism groups of these Artin groups are shown to be small: order 2 for $B_n$, and order 4 (resp. 2) for $D_n$ when $n$ is even (resp. odd).

ABSTRACT

We show that each of the Artin groups of type $B_n$ and $D_n$ can be presented as a semidirect product $F times {\cal B}_n$, where $F$ is a free group and ${\cal B}_n$ is the $n$-string braid group. We explain how these semidirect product structures arise quite naturally from fibrations, and observe that, in each case, the action of the braid group ${\cal B}_n$ on the free group $F$ is classical. We prove that, for each of the semidirect products, the group of automorphisms which leave invariant the normal subgroup $F$ is small: namely, ${ m Out}(A(B_n),F)$ has order 2, and ${ m Out}(A(D_n),F)$ has order 4 if $n$ is even and 2 if $n$ is odd. It is known that the Artin group of type $D_n$ may be viewed as an index 2 subgroup of the $n$-string braid group over some orbifold. Applying the same techniques, we show that this latter group has an outer automorphism group of order 2. Finally, we determine the automorphism groups of all Artin groups or rank 2.

Motivation & Objective

• To understand the internal structure of Artin groups of type $B_n$ and $D_n$ by expressing them as semidirect products involving the braid group $\mathcal{B}_n$.
• To explain the geometric origin of these semidirect product structures via fibrations of regular orbit spaces over $N(A_{n-1})$.
• To determine the outer automorphism groups of $A(B_n)$ and $A(D_n)$, particularly the size of $\mathrm{Out}(A(B_n), F)$ and $\mathrm{Out}(A(D_n), F)$.
• To compute the full automorphism groups of Artin groups of rank 2, especially for types $B_2$ and $D_2$.
• To extend techniques from braid group automorphisms to related groups, including the orbifold braid group associated with $D_n$.

Proposed method

• Presenting $A(B_n)$ and $A(D_n)$ as semidirect products $F \rtimes \mathcal{B}_n$, where $F$ is a free group and $\mathcal{B}_n$ acts on $F$ via classical monodromy actions.
• Using topological fibrations of the regular orbit spaces $N(B_n)$ and $N(D_n)$ over $N(A_{n-1})$ to derive the semidirect product decomposition.
• Analyzing the action of $\mathcal{B}_n$ on $F$ by showing it corresponds to Artin's classical representation for $B_n$ and the monodromy on the Milnor fibre for $D_n$.
• Applying group-theoretic techniques involving conjugacy classes and central elements to classify automorphisms up to inner automorphisms.
• Reducing automorphism problems to the quotient group $\overline{A} = A / Z(A)$, where $Z(A)$ is the center of the Artin group.
• Using presentations of rank-2 Artin groups in terms of $a, b$ with relations $a^m = b^2$ (for $m$ odd) or $a^k b = b a^k$ (for $m$ even) to analyze automorphisms via induced maps on $\overline{A}$.

Experimental results

Research questions

• RQ1 How can the Artin groups of type $B_n$ and $D_n$ be decomposed as semidirect products involving the braid group $\mathcal{B}_n$?
• RQ2 What is the geometric or topological origin of the semidirect product structure in $A(B_n)$ and $A(D_n)$?
• RQ3 What is the size of the outer automorphism group $\mathrm{Out}(A(B_n), F)$, where $F$ is the normal free subgroup?
• RQ4 What is the size of $\mathrm{Out}(A(D_n), F)$, and how does it depend on the parity of $n$?
• RQ5 What are the full automorphism groups of rank-2 Artin groups of type $B_2$ and $D_2$?

Key findings

• The Artin group $A(B_n)$ admits a semidirect product decomposition $F \rtimes \mathcal{B}_n$, where $F$ is a free group of rank $n-1$, and the action of $\mathcal{B}_n$ on $F$ is Artin's classical representation.
• The Artin group $A(D_n)$ also admits a semidirect product structure $F \rtimes \mathcal{B}_n$, with the action arising from the monodromy on the Milnor fibre of an $A_{n-1}$-type singularity.
• The outer automorphism group $\mathrm{Out}(A(B_n), F)$ has order 2, meaning only two automorphisms preserve the normal subgroup $F$ up to inner automorphisms.
• For $A(D_n)$, the outer automorphism group $\mathrm{Out}(A(D_n), F)$ has order 4 if $n$ is even and order 2 if $n$ is odd.
• The automorphism group of the rank-2 Artin group of type $B_2$ is trivial after factoring out inner automorphisms, as shown by the proof that only the identity automorphism satisfies the required conditions.
• The outer automorphism group of the orbifold braid group associated with $D_n$ is of order 2, extending the result to a related group structure.