Symbolic dynamics for Kleinian groups
This thesis is composed of three independent chapters and an appendix. In the first two chapters we deal with Kleinian groups and in the third one we concentrate on Fuchsian groups. In chapters 1 and 2, we study the action of a Kleinian group on points in hyperbolic space of three dimensions and on points on its boundary. All Kleinian groups we study have the property that their action on points in the hyperbolic 3-space has a fundamental polyhedron with the property that the union of the images of its faces under the group elements contains each geodesic plane that contains one of the faces. This is the so called even corners property. Using this property we prove in Chapter one the existence of an expanding Markov type mapping on the Riemann sphere. The main application is to give a shorter proof that the Selberg zeta function associated to the Kleinian group has a meromorphic extension to C. In chapter two, we assume the existence of a Kleinian group with the even corner property and define its deformation space. We then prove that the Hausdorff dimension of the limit set of groups in the deformation space varies real analytically as we vary the points in the deformation space. We prove that Klein's combination Theorem quasi-preserves the even corners property in the sense that, if a group r is formed by other two via Klein's theorem and these last two are quasi-conformal deformations of groups with the even corners property, then r is a deformation of a Kleinian group with the even corners property. In particular, the result about the Hausdorff dimension is valid for all geometrically finite purely loxodromic function groups. In chapter three we construct an automatic structure for parabolic free Fuchsian groups based on the symbolic coding of points in their limit sets. We then provide a proof, based on their symbolic dynamics, that these groups are automatic. We explicitly determine an automatic structure for the groups.