In this paper, we consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We then construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. As an application, we will affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks.

In this paper, we provide new discrete uniformization theorems for bounded, m-connected planar domains. To this end, we consider a planar, bounded, m-connected domain Ω and let Ω be its boundary. Let 𝒯 denote a triangulation of Ω∪Ω. We construct a new decomposition of Ω∪Ω into a finite union of quadrilaterals with disjoint interiors. The construction is based on utilizing a pair of harmonic functions on 𝒯^(0) and properties of their level curves. In the sequel it will be proved that a particular discrete scheme based on these theorems converges to a conformal map, thus providing an affirmative answer to a question raised by Stephenson .

We show that geometrically finite Kleinian groups are dense in the boundary of the quasiconformal deformation space of any geometrically finite Kleinian group.

We study the growth of fibers of coverings of pinched negatively curved Riemannian manifolds. The applications include counting estimates for horoballs in the universal cover of geometrically finite manifolds with cusps. Continuing our work on diophantine approximation in negatively curved manifolds started in an earlier paper (Math. Zeit.241 (2002), 181–226), we prove a Khintchine–Sullivan-type theorem giving the Hausdorff measure of the geodesic lines starting from a cusp that are well approximated by the cusp returning ones.

Let M be a compact, oriented, irreducible, atoroidal 3-manifold with nonempty boundary. Let CC0(M ) denote the space of convex cocompact Kleinian groups uniformizing M . We show that any Kleinian group in the boundary of CC0(M ) whose limit set is the whole sphere can be approximated by maximal cusps. Density of maximal cusps on the boundary of Schottky space is derived as a corollary. We further show that maximal cusps are dense in the boundary of the quasiconformal deformation space of any geometrically finite hyperbolic 3-manifold with connected conformal boundary.

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. We study the asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M. In the appendix, due to K. Belabas, the case of SL(2,Z) and of Bianchi groups is developed.

We prove an ergodic rigidity theorem for discrete isometry groups of CAT(-1) spaces.We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2-polyhedra, hyperbolic Bruhat-Tits buildings and rank one symmetric spaces.We prove that two negatively curved Riemannian metrics, with conical singularities of angles at least 2π, on a closed surface, with boundary map absolutely continuous with respect to the Patterson-Sullivan measures, are isometric.For that, we generalize J.-P.Otal's result to prove that a negatively curved Riemannian metric, with conical singularities of angles at least 2π, on a closed surface, is determined, up to isometry, by its marked length spectrum.

Let $T$ be a locally finite simplicial tree and let $\Gamma\subset{\rm Aut}(T)$ be a finitely generated discrete subgroup. We obtain an explicit formula for the critical exponent of the Poincaré series associated with $\Gamma$, which is also the Hausdorff dimension of the limit set of $\Gamma$; this uses a description due to Lubotzky of an appropriate fundamental domain for finite index torsion-free subgroups of $\Gamma$. Coornaert, generalizing work of Sullivan, showed that the limit set is of finite positive measure in its dimension; we give a new proof of this result. Finally, we show that the critical exponent is locally constant on the space of deformations of $\Gamma$.

We give a generalization of the Shimizu-Leutbecher inequality and a partial generalization of the Jorgensen inequality to Möbius transformations in R N {{\mathbf {R}}^N} using the Clifford algebra and the Vahlen group.