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

Expanding Thurston Maps

We study the dynamics of Thurston maps under iteration. These are branched covering maps $f$ of 2-spheres $S^2$ with a finite set $\mathop{post}(f)$ of postcritical points. We also assume that the maps are expanding in a suitable sense. Every expanding Thurston map $f\: S^2 \to S^2$ gives rise to a type of fractal geometry on the underlying sphere $S^2$. This geometry is represented by a class of \emph{visual metrics} $\varrho$ that are associated with the map. Many dynamical properties of the map are encoded in the geometry of the corresponding {\em visual sphere}, meaning $S^2$ equipped with a visual metric $\varrho$. For example, we will see that an expanding Thurston map is topologically conjugate to a rational map if and only if $(S^2, \varrho)$ is quasisymmetrically equivalent to the Riemann sphere $\widehat{\mathbf{C}}$. We also obtain existence and uniqueness results for $f$-invariant Jordan curves $\mathcal{C}\subset S^2$ containing the set $\mathop{post}(f)$. Furthermore, we obtain several characterizations of Latt\`{e}s maps.Comment: 492 pages, 51 figure

Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G 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 hyperbolic 3-space.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper7.abs.htm

Journal of African Christian Biography: v. 1, no. 1

A publication of the Dictionary of African Christian Biography with U.S. offices located at the Center for Global Christianity and Mission at Boston University. This issue focuses on: 1. "Walatta Petros and Hakalla Amale, Pious Women of Ethiopia," with commentary by Dr. Jonathan Bonk, Project Director. 2. Walatta Petros. 3. Hakalla Amale. 4. Recent Print and Digital Resources Related to Christianity in Africa