This project is related to "Alice on the Eightfold-Way". The plan is to visualize arbitrary hyperbolic 3-manifolds in the CUBE. More specifically, the goal is to allow users to walk around in manifolds given in SnapPea's triangulation file format. A virtually unlimited number of manifolds in this format is already available (see, for example, the Hodgson & Weeks census and Peter Brinkmann's train track software).
One of the fundamental problems in group theory is the question of whether two elements of a group are conjugate. The paper math.GT/0012001 contains an effective new solution of the conjugacy problem in the mapping class group of surfaces of one puncture. This solution consists of necessary and sufficient conditions for conjugacy. An implementation of the sufficient condition is already available, and the goal of this REU project is to implement the necessary condition.
The goal of this project is to design and implement massive computer experiments in order to investigate possible relationships between the growth rate of pseudo-Anosov homeomorphisms of surfaces and the hyperbolic volume of their mapping tori. Similar experiments involving the volume of certain knot complements have given rise to rather striking results and interesting (and difficult) questions.