Commensurability classes of three and four dimensional hyperbolic simplices
The main objects of study in this thesis are the reflection groups associated with hyperbolic simplices. Initially we will restrict ourselves to three dimensions, and in particular to the non-compact cases which arise. The compact cases have been studied in detail by Reid. We will use the upper half-space model of hyperbolic three space to obtain representations of these groups as Klienian groups. Then by using established techniques for determining the arithmeticity and commensurability of Klienian groups, we will divide them accordingly into commensurability classes. Profiting by some existing ideas, we will then establish geometric results which be used to confirm explicitly the results on commensurability. In the latter part of the thesis we move up a dimension to hyperbolic four space and discuss the simplex groups there and, in particular, their existence through Coxeter schemes and Gram matrices. Their arithmeticity is also established and, for completeness, a calculation of their volumes. These results are derived mainly from the work of Vinberg. We will then produce a test for commensurability from ideas of Borel and Harish-Chandra. It uses quadratic spaces, and the one-to-one correspondence between classes of quadratic forms and commensurability classes of orthogonal groups. The test developed for classifying the simplex groups is original but uses derived tools and ideas from various sources. As all the hyperbolic four space simplices are arithmetic, this classification by associated quadratic forms into commensurability classes is complete. Some of these results have been derived by other authors using different techniques. Again geometric results are established in order to confirm the algebraic results. Finally there is a discussion of how these ideas could be extended into higher dimensions and to other objects in hyperbolic space.