Geometric invariants of spaces with isolated flats

We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats and the Relative Fellow Traveller Property. The groups in question include word hyperbolic CAT(0) groups as well as geometrically finite Kleinian groups and numerous 2-dimensional CAT(0) groups. For such a group we show that there is an intrinsic notion of a quasiconvex subgroup which is equivalent to the inclusion being a quasi-isometric embedding. We also show that the visual boundary of the CAT(0) space is actually an invariant of the group. More generally we show that each quasiconvex subgroup of such a group has a canonical limit set which is independent of the choice of overgroup. The main results in this article were established by Gromov and Short in the word hyperbolic setting and do not extend to arbitrary CAT(0) groups.Comment: 21 pages, 2 figures. Submitte

The Medusa Algorithm for Polynomial Matings

The Medusa algorithm takes as input two postcritically finite quadratic polynomials and outputs the quadratic rational map which is the mating of the two polynomials (if it exists). Specifically, the output is a sequence of approximations for the parameters of the rational map, as well as an image of its Julia set. Whether these approximations converge is answered using Thurston's topological characterization of rational maps. This algorithm was designed by John Hamal Hubbard, and implemented in 1998 by Christian Henriksen and REU students David Farris, and Kuon Ju Liu. In this paper we describe the algorithm and its implementation, discuss some output from the program (including many pictures) and related questions. Specifically, we include images and a discussion for some shared matings, Lattes examples, and tuning sequences of matings.Comment: 25 pages, many figures, submitte