Sa’ar Hersonsky

Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions

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 theHausdorffmeasureofthegeodesiclines startingfromacuspthatarewell approximated by the cusp returning ones.

Boundary value problems on planar graphs and flat surfaces with integer cone singularities, I: The Dirichlet problem

Abstract Consider a planar, bounded, m-connected region Ω, and let ∂Ω be its boundary. Let 𝒯 be a cellular decomposition of Ω ∪ ∂Ω, 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 flat surface tiled by rectangles and f is an energy preserving mapping from 𝒯(1) onto S. By a singular flat surface, we will mean a surface which carries a metric structure locally modeled on the Euclidean plane, except at a finite number of points. These points have cone singularities, and the cone angle is allowed to take any positive value (see for instance [28] for an excellent survey). Our realization may be considered as a discrete uniformization of planar bounded regions.

Counting Horoballs and Rational Geodesics

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 groups, is developed.

Energy and length in a topological planar quadrilateral

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.

Diophantine Approximation in Negatively Curved Manifolds and in the Heisenberg Group

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-Caratheodory metric, can be seen as the space at infinity of the complex hyperbolic space (minus one point). The rational approximation on the Heisenberg group can be interpreted and developed using arithmetic subgroups of SU (n, 1). In the appendix, the case of hyperbolic surfaces is developed by Jouni Parkkonen and the second author.

Groups of automorphisms of trees and their limit sets

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 Poincare 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 index torsion-free subgroups of !. 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 !.

The first betti number of the smallest closed hyperbolic 3-manifold

It follows from the work of Gromov, Jorgensen and Thurston (see [3]) that the real numbers which arise as volumes of hyperbolic 3-manifolds form a well-ordered set. It is not known at present which closed 3-manifold has the minimal volume (or whether such a manifold is unique). The techniques developed in the series of papers [6-9,1], bear on this question since they give volume estimates which depend on topological properties of the manifold. If a certain topological hypothesis can be shown to imply a volume bound that exceeds the volume of a known manifold, one obtains topological information about any minimal volume manifold. The first estimates to have interesting qualitative consequences of this sort appeared in [l]. In the present paper we prove the following result.

Approximation of conformal mappings and novel applications to shape recognition of planar domains

Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint—a computable approximation to the conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22 for a few examples). In this paper, we make the first significant step in this program where we address the case of simply and doubly connected planar domains. We prove uniform convergence of our approximation scheme to the appropriate conformal mapping. To this end, we affirm a conjecture raised by Ken Stephenson in the 1990s 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 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. The resolution of Stephenson’s conjecture then follows by a limiting argument. With more complex topology of the given shape, i.e., when it has higher genus, we will use methods invented by Arabnia (J Parallel Distrib Comput 10:188–192, 1990) and Wani–Arabnia (J Supercomput 25:43–62, 2003). First, to divide the domain into subdomains and thereafter to make the scheme presented in this paper suitable for parallel processing. We will then be able to compare our results for those appearing, for instance, in the work of Arabnia–Oliver (Comput Graph Forum 8:3–11, 1989) that provides algorithms for the translation and scaling of complicated digitalized images.