Growth,Almost Convexity, and Automatic Structure in Some Geometric Three-Manifold Groups

This thesis grows out of the attempt to understand the fundamental groups of three-dimensional manifolds which carry one of Thurston's eight geometries. In particular, we study the fundamental groups of manifolds which have the Nil geometry, or the PSL(2,R) geometry. These groups are central Z extensions of surface groups. We show that in certain cases, we can exhibit a central Z extension, G, as equivalence classes of based edge paths in the Cayley graph of the quotient group, H. Given appropriate generating sets, this isomorphism is geodesic in the sense that the length of a shortest path in the equivalence class is the length of the element it represents. We then extend this method to the case where G does not appear as such a path group by showing that it may be understood as made up of "satellites" of a group which does. Nil and PSL(2,R) groups have well understood quotient groups, namely, the torus group, and hyperbolic surface groups. Using geometric properties of the Cayley graphs of the quotient groups, we are able to develop detailed geodesic information about the Cayley graphs of Nil and PSL(2,R) groups. We compute the growth function of an arbitrary discrete, co-compact Nil group, and show that these groups are almost convex. We show that some PSL(2,R) groups have rational growth and automatic structure. We show that others are almost convex, and we are able to develop local information about growth which depends strikingly on choice of generating set.