Cocycle growth for the Steinberg representation

This thesis investigates the growth of the natural cocycle introduced by Klingler for the Steinberg representation. When possible, we extend the framework of simple algebraic groups over a local field to arbitrary Euclidean buildings. In rank one, the growth of the cocycle is determined to be sublinear. In higher rank, the complexity of the problem leads us to study of the geometry of buildings of dimension two, where we describe in details the relative position of three points.