Rigidity and Stability for Isometry Groups in Hyperbolic 4-Space by Youngju Kim Advisor: Professor Ara Basmajian It is known that a geometrically finite Kleinian group is quasiconformally stable. We prove that this quasiconformal stability cannot be generalized in 4-dimensional hyperbolic space. This is due to the presence of screw parabolic isometries in dimension 4. These isometries are topol...

Given a complex space X, we cosidered the problem of finding a hyperbolic model of X. This is an object Ip(X) with a morphism i : X → Ip(X) in such a way that Ip(X) is “hyperbolic” in a suitable sense and i is as close as possible to be an isomorphism. Using the theory of model categories, we found a definition of hyperbolic simplicial sheaf (for the strong topology) that extends the classical ...

In this paper, we give an application of iterated pushdown automata to contour words of balls and two other domains in infinitely many tilings of the hyperbolic plane. We also give a similar application for the tiling {5, 3, 4} of the hyperbolic 3D space and for the tiling {5, 3, 3, 4} of the hyperbolic 4D space as well. keywords: pushdown automata, iterated pushdown automata, tilings, hyperbol...

We prove that the space of SU(2) hyperbolic monopoles based at the centre of hyperbolic space is homeomorphic to the space of (un-based) rational maps of the two-sphere. The homeomorphism extends to a map of the natural compactiications of the two spaces. We also show that the scattering methods used in the study of monopoles apply to the connguration space for hyperbolic monopoles giving a hom...

Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of infinity. We call a C1 endomorphism of the compact interval (or the circle) hyperbolic if it has finitely many hyperbolic attracting periodic points and the complement of the basin of attraction of these poi...

We prove asymptotic completeness in the energy space for the nonlinear Schrödinger equation posed on hyperbolic space Hn in the radial case, for n > 4, and any energy-subcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which kind of interpolate between Euclidean space and hyper...

Internet coordinate systems have emerged as an efficient method to estimate the latency between pairs of nodes without any communication between them. They avoid the cost of explicit measurements by placing each node in a finite coordinate space and estimating the latency between two nodes as the distance between their positions in the space. In this paper, we adapt the Vivaldi algorithm to use...

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial ...