There are several sets of axioms for both Euclidean and hyperbolic plane geometry. If the axioms in each such set are combined, we could say that we have definitions of Euclidean or hyperbolic planes. My goal is to describe one formulation and how the uniqueness theorems can be proved in that context. In this formulation, the real numbers are taken as a starting point, rather than deriving the real numbers from other properties of planes. The two uniqueness theorems have a number of steps in common, consisting of what are called neutral theorems, i.e., ones that do not depend on any assumption about the number of lines parallel to a given line through a given point. Indeed one of the theorems is that there is always at least one such parallel line. I will only mention some of the theorems more central to the development.
Many finite Coxeter groups occur naturally as groups of automorphisms of polytopes (higher dimensional polyhedra). Some of my recent work considers subcomplexes of these polytopes, that is, spaces obtained by deleting interiors of certain faces. If these subcomplexes are chosen carefully, their homology groups (don't worry, I'll explain this) afford representations of the Coxeter group. I will discuss how one might find the characters of such representations, concentrating mostly on the case of type D. This will be a good excuse to discuss the representation theory of finite Coxeter groups of classical type (A, B, D) over the complex numbers.
