Mireille Soergel - Dyer groups Coxeter groups right-angled Artin groups and more...

Beschreibung

Talk

This is exactly the same talk as in Mireille Soergel - Dyer groups Coxeter groups right-angled Artin groups and more. We define the Right-angled Artin group by a finite simplicial (?) graph. We will study the word problem on those groups. i.e. is a word the identity? $\Gamma$

We define two $M$-transformation, which reduces and keeps lengths of words of a coxeter groups.

We define the coxeter group from a graph. Its generating set is the vertex set. All generators have order 2. We have a relation $(uvuvuv...)$ and $(uvuvuvuvu...)$ with $m$ repetitions if the Graph has an edge between the vertices with label $m$.
We ask the same word problem question for Artin groups. This time we need to change up the second transformation.

We ask, is there some generalisation of Coxeter and Artin groups where the word problem can be solved in a analogous manner? Yes! The Dyer groups!

Definition

A Dyer group is formed from a finite simplicial graph. We have labelled the graph with vertices and labels (with some restrictions). We then form a group. The vertices are the generator. The labels of the vertices tell us the order of the generators the labels on the edges tell us something on the triviality of $uvuvuvu$ repetitions.

This theory is not related to Bass-Serre theory?

Those Dyer groups include Coxeter group, Right-angled Artin groups. Are those groups CAT(0)? Groups are called CAT(0) if they act freely, properly discontinuously on a CAT(0) space. Yes! (S. 2024)

They are finite index subgroup of Coxeter groups. And Coxeter groups are CAT(0).
The idea of the proof here is to tweak the diagrams of Dyer groups to get coxeter groups.

But this is not a satisfying proof. It doesn’t give a space the group acts on. We want to find a Cat(0) simplicial complex such that it is the Davis-Mousseng complex for Coxeter groups and the Salvetti complex for RAAGs. We let a group act on a complex. Just like in Bass-Serre theory this tells us something on the groups. To find the simplicial complex we find a suitable quotient complex and reconstruct the complex from that. We look at the Dyer graph and try to find subgraph which generate groups we understand. This is made by taking subsets of groups but only those which generate “nice” groups. We create a graph out of those generators. The idea here is that the generators are exactly the stabilizers of the points labelled by the generators. We first create the simplicial complex topologically. Then we define a metric. This metric will be important in distinguishing the $m$ on the edge labels. (Because we want the universal cover (the opposite of the quotient space) to have angle sum $2\pi$)

References

• Green (1990)
• Tits (1969)