Abstract

Recall that an Artin group is called of spherical type if its associated Coxeter group is finite. It is known that the irreducible spherical Artin groups consist of four infinite families, the groups of type $A_{n}$ , the groups of type $B_{n}$ , the groups of type $D_{n}$ and the groups of type $I_{2} (m)$ , and 6 sporadic groups. The groups of type $A_{n}$ are the braid groups and they can be easily understood using various algebraic and/or topological methods. The groups of type $B_{n}$ are finite index subgroups of the groups of type $A_{n}$ , hence the methods used to understand the groups of type An often apply in this case as well. The groups of type $I_{2} (m)$ are virtually direct products of a free group with $Z$ . However, in order to understand the groups of type $D_{n}$ , it is often necessary to develop new tools and they are more difficult to study. This talk will recount the different studies on these groups to culminate in the latest one, in collaboration with Fabrice Castel, where we classify the endomorphisms of such groups for a large $n$ .

Talk

The author presents a special representation which might be interesting.

Introduction

Where do Artin group come from?

A Coxeter matrix is a matrix with one one the diagonal and numbers bigger than 1 in all other entries.
We create a graph called the Coxeter graph with an edge if the entry is bigger equals $3$ .
$Π (a, b, m)$ is the alternating word $abababab ...$ with $m$ letters.
We define the Artin group $A [Γ]$ which has the vertices as generators and the generators $Π (s, t, m_{s, t}) = Π (t, s, m_{s, t})$ . The Coxeter Group $W [Γ]$ is created by taking the quotient by $s^{2} = 1$ for all generators $s$ .
We say the two groups are from spherical type if $W [Γ]$ is finite
We call them irreducible if $W [Γ]$ is ???
For the graph being a line the Artin group becomes the Braid group.

The goal of the talk is to study groups of the graph consisting of a line with a short split at the end.

Monodromy representation

The same way we can represent the braid group with standard generators and relators we can do the same with Artin groups and get the Generalized Artin representation. But instead of free elements we map to the Automorphism group of products of cyclic $C_{2}$ groups. Crisp-Paris 2005 shows this is faithful.

Some extra steps then give us the so-called Monodromy representation . Perron-Vannier 1998 and Crisp-Paris 2005 showed that this is faithful too.

Next we look again at the line graph and the forked line graph. The graph map merging the fork induces a map on the Artin groups. This is an epimorphism and it admits a section (whatever that is) but it shows that the Artin group of the forked line $A [D_{n}]$ can be written as a Semi-direct product (Perron-Vannier and Crisp-Paris 2005).

Geometric monodromy

We consider a compact oriented surface $Σ$ with or without boundary and a finite collection of punctures. We study the group of orientation-preserving homeomorphism, preserving the punctures setwise and acting as the identity on the boundary.

We study the Mapping class group of the group. We define a Essential simple closed curve. Two disjoint circles induce commutative Dehn twists, if it is one point they fulfill a relation, similar to the Artin groups (compare halftwist on the $n$ -punctured disc).

We take a special surface with $n /2$ -holes (not punctures) such that we have $n$ essential simple closed curves. There is a homeomorphism mapping braid generators to Dehn-twist. This is injective and called the Geometric monodromy. If I understand corretlz by Dehn-Nielsen-Baer this is faithful.

Topological interpretation

For a Coxeter graph of spherical type, we can always find a linear canonical representation. A theorem in Breiskorn 1972 and Crisp-Paris 2005) gives us some connection to the Classifying space. And a map is defined such that the homeomorphism induces a fiber-bundle on the classifying space.

Birman exact sequence

Applications

We can classify the outer automorphism group

About

Long introduction
Jokes in introduction
Explained the goal very clearly
Anticipates what the viewer doesn’t know
21 pages

🪴 Quartz 4.0

Explorer

Luis Paris - Artin groups of type Dn