Abstract: 

Coxeter complexes arise as highly symmetric simplicial structures associated with Coxeter groups, encoding both algebraic and geometric information. They play a central role in the study of buildings, where global geometric properties can often be reduced to local combinatorial questions within a single complex. In this talk, we introduce Coxeter groups and their associated complexes, explain how buildings can be understood as unions of these complexes, and how retractions allow us to study global behaviour via a fixed Coxeter complex. We will then introduce a new concept called the annex, which is the complement to a Bruhat interval in the Coxeter group. We will explore some properties of these sets and see that, in the case that the group is affine, these sets are finite. Lastly, in rank 2 we will see a description of the boundary of the annex.