| Isometric orbifolds have isomorphic fundamental groups. Deformation theory attempts to explain how close the converse is to being true. That is, deformation theory investigates the different isometric structures an orbifold can have while maintaining the same fundamental group. The Mostow-Prasad rigidity theorem states that if two 3-orbifolds with finite volume hyperbolic structures have isomorphic fundamental groups, then they are isometric. In contrast, a surface of genus g has a 6g - 6 smooth variety of isometric structures.;This work begins to explain when a lattice admits a deformation and then, when it does, to classify the types of deformation that occur in some systematic way. The first step is to find the dimension of the deformation space of a large class of cofinite Coxeter groups. Then we develop tools to analyze the deformation space of groups commensurable with cofinite Coxeter groups. Using these tools, we are able to describe the deformation spaces of Bianchi groups which have a finite index Coxeter subgroup. Further, we describe the structure of some small dimensional deformation spaces by exploiting the structure of Coxeter groups. We are also able to exhibit explicitly the two dimensional deformation spaces of the Lambert cubes and show that it does not contain any bending deformation. Finally, we are able to give the first proof that the "stamping example" of Apanasov was the first explicit deformation in print which is independent of bends. |
