Mario Bonk

Embeddings of Gromov Hyperbolic Spaces

It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant.

Quasisymmetric parametrizations of two-dimensional metric spheres

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standard 2-sphere, answering a question of Heinonen and Semmes.

Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary

Suppose G is a Gromov hyperbolic group, and @1G is quasisymmetrically homeomorphic to an Ahlfors Q–regular metric 2–sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on H 3 .

Expanding Thurston Maps

We study the dynamics of Thurston maps under iteration. These are branched covering maps

Uniformization of Sierpiński carpets in the plane

f

Quasi-hyperbolic planes in hyperbolic groups

of 2-spheres

Logarithmic potentials, quasiconformal flows, and

S^2

Q

with a finite set

-curvature

\mathop{post}(f)

Covering properties of meromorphic functions, negative curvature and spherical geometry

of postcritical points. We also assume that the maps are expanding in a suitable sense. Every expanding Thurston map