On Brady's Classifying Spaces for Artin Groups of Finite Type

TL;DR

This paper generalizes Brady's construction of classifying spaces for braid groups to all Artin groups of finite type, establishing their $K( ext{pi},1)$ spaces using non-crossing partitions and poset groups.

Contribution

It extends Brady's $K( ext{pi},1)$ construction from braid groups to all finite type Artin groups using non-crossing partitions and poset group techniques.

Findings

Constructed a simplicial complex serving as the universal cover of the $K( ext{pi},1)$ for finite type Artin groups.

Proved the poset group is isomorphic to the Artin group.

Demonstrated the quotient of the complex yields the $K( ext{pi},1)$ space.

Abstract

This thesis takes Brady's construction of $K(\pi,1)$s for the braid groups as a starting point. It is widely known that this construction can - with the right ingredients - be generalized to Artin groups of finite type. Results of Bessis as well as Brady and Watt are used to establish the general construction for Artin groups of finite type. The non-crossing partition lattice in finite Coxeter groups is identified and used to generate the so called poset group. With the help of this poset group, which turns out to be isomorphic to the Artin group, a simplicial complex is constructed on which the poset group acts. It is shown that the complex itself is the universal cover of the $K(\pi,1)$ of the Artin group of finite type and the quotient under the action is the desired $K(\pi,1)$.