A KLEINIAN APPROACH TO FUNDAMENTAL REGIONS

This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\ obius transformations, have special properties and turn out to be linear fractional transformations when in $U^2$ and $B^2$. We will then study a bit of topology, specifically the topological groups relevant to the group of isometries of hyperbolic $n$-space, $I(H^n)$. Finally we will combine what we know about hyperbolic $n$-space and topological groups in order to study fundamental regions, fundamental domains, Dirichlet domains, and quotient spaces. Using examples in $U^2$, we will then illustrate how useful fundamental domains are when it comes to visualizing the geometry of quotient spaces