Dihedral Branches Coverings of Hyperbolic Orbifolds

We consider the class of hyperbolic 3-orbifolds whose underlying topological space is the 3-sphere S ³ and whose singular set is a trivalent graph with singular index 2 along each edge (an important special case occurs when the trivalent graph is the 1-skeleton of a hyperbolic polyhedron). Our main result is a classification of the D-branched coverings of these orbifolds (where D₂ is the dihedral group of order 4) under some general conditions on their isometry groups or the lengths of their geodesics.     

Minimal length of two intersecting simple closed geodesics

On a hyperbolic Riemann surface, given two simple closed geodesics that intersect n times, we address the question of a sharp lower bound L _n on the length attained by the longest of the two geodesics. We show the existence of a surface S _n on which there exists two simple closed geodesics of length L _n intersecting n times and explicitly find L _n for . The first author was supported in part by SNFS grant number 2100-065270, the second author was supported by SNFS grant number PBEL2-106180.     

Dégénérescence de séries d’Eisenstein hyperboliques

In this work, we deal with hyperbolic Eisenstein series and more particularly, given a degenerating family S _l (l ≥ 0) of Riemann Surfaces with their canonical hyperbolic metrics, we work out the degeneration of hyperbolic Eisenstein series associated to the pinching geodesics in S _l . Our principal results are theorems 2.2, 4.1 et 4.2.     

Geodesics in large planar maps and in the Brownian map

We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of S\{x}. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps.     

Legendrian Links, Causality, and the Low Conjecture

Let (X ^m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of ${\mathbb{R}^{m}}Let (X ^m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of \mathbbR^m{\mathbb{R}^{m}} . The Legendrian Low conjecture formulated by Natário and Tod says that two events x, y ? X{x, y \in X} are causally related if and only if the Legendrian link of spheres \mathfrakS_x, \mathfrakS_y{{\mathfrak{S}_x,\,\mathfrak{S}_y}} whose points are light geodesics passing through x and y is non-trivial in the contact manifold of all light geodesics in X. The Low conjecture says that for m = 2 the events x, y are causally related if and only if \mathfrakS_x, \mathfrakS_y{{\mathfrak{S}_x,\,\mathfrak{S}_y}} is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic (X ^m+1, g) such that a cover of its Cauchy surface is diffeomorphic to an open domain in \mathbbR^m{\mathbb{R}^{m}} .     

On a multiplicity one property for the length spectra of even dimensional compact hyperbolic spaces

We prove a multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces, i.e., if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.     

An arithmetic property of the set of angles between closed geodesics on hyperbolic surfaces of finite type

For a hyperbolic surface S of finite type we consider the set A(S) of angles between closed geodesics on S. Our main result is that there are only finitely many rational multiples of \(\pi \) in A(S).     

Projection theorems in hyperbolic space

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of the hyperbolic n-space by a geometric argument. Moreover, we obtain a Besicovitch–Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections.

     

The Moduli Space of Complex Geodesics in the Complex Hyperbolic Plane

We consider the space M\mathcal{M} of ordered m-tuples of distinct complex geodesics in complex hyperbolic 2-space, H_\mathbbC²{\rm\bf H}_{\mathbb{C}}^{2}, up to its holomorphic isometry group PU(2,1). One of the important problems in complex hyperbolic geometry is to construct and describe the moduli space for M\mathcal{M}. This is motivated by the study of the deformation space of groups generated by reflections in complex geodesics. In the present paper, we give the complete solution to this problem.     

Simple Curves on Surfaces

We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L ^6g+2b+2c–6. This answers a long-standing open question.     

Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l.     

Geodesic Laminations Revisited

The Bratteli diagram is an infinite graph which reflects the structure of projections in an AF-algebra. We prove that every strictly ergodic unimodular Bratteli diagram of rank 2g+m−1 gives rise to a minimal geodesic lamination with the m principal regions on a hyperbolic surface of genus g≥1. The proof is based on a Morse theory of the recurrent geodesics on the hyperbolic surfaces.     

Growth Series and Random Walks on Some Hyperbolic Graphs

 Consider the tessellation of the hyperbolic plane by m-gons, ℓ per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of proper loops in these graphs. We then apply Grigorchuk’s result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs.     

Mapping Class Groups of Hyperbolic Surfaces and Automorphism Groups of Graphs

Let M be a hyperbolic surface and (M) its extended mapping class group. We show that (M) is isomorphic to the automorphism group of the following graph G(M). The set of vertices of G(M) is the set S(M) of nonseparating simple closed geodesics of M. Two vertices u and v of S(M) are related by an edge if u and v intersect exactly once in M. The graph G(M) can be thought of as a combinatorial model for M.     

Growth Series and Random Walks on Some Hyperbolic Graphs

 Consider the tessellation of the hyperbolic plane by m-gons, ℓ per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of proper loops in these graphs. We then apply Grigorchuk’s result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs. Received 19 September 2001; in revised form 23 December 2001     

Hyperbolic Mapping Classes and Their Lifts on the Bers Fiber Space

Let S be a Riemann surface with genus p and n punctures. Assume that 3p - 3 n ＞ 0 and n ≥ 1. Let a be a puncture of S and let (～S) = S ∪ {a}. Then all mapping classes in the mapping class group Mods that fixes the puncture a can be projected to mapping classes of Mod(～S) under the forgetful map. In this paper the author studies the mapping classes in Mods that can be projected to a given hyperbolic mapping class in Mod(～S).     

The existence of two closed geodesics on every Finsler 2-sphere

In this paper, we prove that for every Finsler metric on S ² there exist at least two distinct prime closed geodesics.     

Stable windings on hyperbolic surfaces

Let ℳ be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law under the Patterson-Sullivan measure on T ¹ℳ of the windings of the geodesics of ℳ around the cusps. This limit law is stable with parameter 2δ− 1, where δ is the Hausdorff dimension of the limit set of the subgroup Γ of M?bius isometries associated with ℳ. The normalization is t ^{−1/(2δ−1)}, for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan measure mentioned above by measures that are regular along the stable leaves. Received: 8 October 1999 / Revised version: 2 June 2000 / Published online: 21 December 2000     

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

K3 surfaces with interesting groups of automorphisms

By the fundamental result of I. I. Piatetsky-Shapiro and I. R. Shafarevich (1971), the automorphism group Aut(X) of aK3 surfaceX over and its action on the Picard latticeS _X are prescribed by the Picard latticeS _X . We use this result and our method (1980) to show the finiteness of the set of Picard latticesS _X of rank 3 such that the automorphism group Aut(X) of theK3 surfaceX has a nontrivial invariant sublatticeS ₀ inS _X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic latticesS ₀, this has been proved by the author before (1980, 1995). Thus we extend these results to negative sublatticesS ₀.We give several examples of Picard latticesS _X with parabolic and negativeS ₀.We also formulate the corresponding finiteness result for reflective hyperbolic lattices of hyperbolic type over purely real algebraic number fields. We give many examples of reflective hyperbolic lattices of the hyperbolic type.These results are important for the theory of Lorentzian Kac-Moody algebras and mirror symmetry.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 45, Algebraic Geometry-8, 1997.