Buildings: An Exposition Detailing Construction and Theorems

In the mid to late twentieth century, Jacques Tits’ work in the area of Lie groups and Lie algebras caused him to develop the construct of a building. Since then the topic has expanded to be viewed from a range of different perspectives and has proven useful in a range of other fields of mathematical research. The topic of buildings brings together several areas of mathematics, including combinatorics, incidence geometry, and Coxeter groups, to name but a few. Buildings are used by many as a vehicle to understanding properties of some of the more complex and unworkable groups that one may wish to understand. This is done through having a building upon which a group can act. In this thesis we will see that the building itself can be considered the fundamental object, and motivate ideas that by taking buildings in different geometries we are able to find new examples of groups. There are a variety of ways from which one may approach the study of buildings, each with its own benefits and shortcomings. This thesis provides an introduction to the topic of buildings showing a geometric based construction with recurring examples.