Polygon Patterns in Affine Building

Abstract

In a tree, the thin triangle property can be rephrased as a decomposition of a line. In this work we look at a generalization of the rephrased version of the thin triangle property in higher rank spaces and describe the associated objects in irreducible affine buildings of rank 2 and 3. The main results of this thesis consist of a complete description of patterns in building associated to SL3, and an exhaustive enumeration of patterns in SL4 in terms of patterns in SL3.