Drawing Limit Sets of Kleinian Groups Using Finite State Automata

Analyticity of Hausdorff dimension of limit sets of Kleinian groups

We show that the Hausdorff dimension of the limit set is a real analytic function on the deformation space of a certain class of convex co-compact Kleinian groups, which includes all convex co-compact function groups. This extends a result of Ruelle [21] for quasifuchsian groups.

On the local connectivity of limit sets of Kleinian groups

AMS no. 30F40 The main result here is the fact that the limit set of a geometrically finite Kleinian group with connected limit set is locally connected. For an analytically finite Kleinian group with connected limit set, we give necessary and sufficient conditions for the limit set to be locally connected.

Wild Knots as limit sets of Kleinian Groups

In this paper we study kleinian groups of Schottky type whose limit set is a wild knot in the sense of Artin and Fox. We show that, if the “original knot” fibers over the circle then the wild knot Λ also fibers over the circle. As a consequence, the universal covering of S − Λ is R. We prove that the complement of a dynamically-defined fibered wild knot can not be a complete hyperbolic 3-manifold.

Calculating Hausdorff Dimension of Julia Sets and Kleinian Limit Sets

Wild Knots in Higher Dimensions as Limit Sets of Kleinian Groups

In this paper we construct infinitely many wild knots, Sn ↪→ Sn+2, for n = 1, 2, 3, 4 and 5, each of which is a limit set of a geometrically finite Kleinian group. We also describe some of their properties.