Saar Hersonsky

University of Georgia | UGA · Department of Mathematics

We extend our discrete uniformization theorems for planar, $m$-connected, Jordan domains [Journal f\"ur die reine und angewandte Mathematik 670 (2012), 65--92] to closed surfaces of non-positive genus.

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, n-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary. 0. Introduction...

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, n-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary. © 2018 Internati...

We 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 fact, we first treat a more general case. Consider a planar annulus, i.e., a bounded, 2-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We construct a cor...

This is the version that was accepted for publication (May 2015) in the Crelle. We extend our discrete uniformization theorems for planar, m-connected, Jor- dan domains [Journal fu ̈r die reine und angewandte Mathematik 670 (2012), 65–92] to closed surfaces of genus m ≥ 1.

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 δΩ be its boundary. Let T denote a triangulation of Ω ∪ δΩ. We construct a new decomposition of Ω ∪ δΩ into a finite union of quadrilaterals with disjoint interiors. The co...

Let T be a triangulation of a closed topological cube Q, and let V be the set of vertices of T. Further assume that the triangulation satisfies a technical condition which we call the triple intersection property (see Definition 3.6). Then there is an essentially unique tiling C = {C v : v ∈ V } of a rectangular parallelepiped R by cubes, such that...

In this paper we continue the study started in part I (posted). We consider a planar, bounded, $m$-connected region $\Omega$, and let $\bord\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\bord\Omega$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we constr...

Consider a planar, bounded, $m$-connected region $\Omega$, and let $\bord\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\bord\Omega$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m-1)$ singular...

Given a negatively curved geodesic metric space $M$, we study the statistical asymptotic penetration behavior of (locally) geodesic lines of $M$ in small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of $M$. We prove Khintchine-type and logarithme law-type results for the spiraling of geodesic lines aro...

Let Γ be a geometrically finite tree lattice. We prove a Khintchine-Sullivan type theorem for the Hausdorff measure of the set of points at infinity of the tree that are well approximated by the parabolic fixed points of Γ. Using Bruhat-Tits trees, an application is given for the Diophantine approximation of formal Laurent series in the variable X...

We provide bounds for the product of the lengths of distinguished shortest paths in a finite network induced by a triangulation of a topological planar quadrilateral.

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 (200...

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M, theHurwi...

This paper is a survey of the work of the authors [21], [2], [22], with a new application to Diophantine approximation in the Heisenberg group. The Heisenberg group, endowed with its Carnot-Carathéodory metric, can be seen as the space at infinity of the complex hyperbolic space (minus one point). The rational approximation on the Heisenberg group...

We provide a sharp estimate for the visual dimension of the set of geodesic rays, starting from any fixed point p in a closed pinched negatively curved Riemannian manifold, that are coming back exponentially close to p infinitely often.

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. 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, are studied in this paper. The case of SL(2, Z), and of Bianchi group...

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 conic...

Let T be a locally finite simplicial tree and let ⊂ Aut(T) be a finitely generated discrete subgroup. We obtain an explicit formula for the critical exponent of the Poincaré series associated with , which is also the Hausdorff dimension of the limit set of ; this uses a description due to Lubotzky of an appropriate fundamental domain for finite ind...

We give a generalization of the Shimizu-Leutbecher inequality and a partial generalization of the Jorgensen inequality to Möbius transformations in RN using the Clifford algebra and the Vahlen group.

We give a generalization of the Shimizu-Leutbecher inequality and a partial generalization of the Jorgensen inequality to Mobius transformations in R(N) using the Clifford algebra and the Vahlen group.