Quasiconformal deformations of complex hyperbolic <Emphasis Type="Italic">p</Emphasis>-Schottky groups

In this paper, we generalize a Schottky group construction to complex hyperbolic space. In particular, we construct a fundamental domain whose sides consist of disjoint non-asymptotic packs for the action of the Schottky group acting on complex hyperbolic space. Then we prove that a smooth deformation of such a Schottky group is quasiconformally stable.     

Dehn filling in relatively hyperbolic groups

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2π Theorem in the context of relatively hyperbolic groups. The first author was supported in part by NSF Grant DMS-0504251. The second author was supported in part by an NSF Mathematical Sciences Post-doctoral Research Fellowship. Both authors thank the NSF for their support. Most of this work was done while both authors were Taussky-Todd Fellows at Caltech.     

p -SOLVABILITY AND A GENERALIZATION OF PRIME GRAPHS OF FINITE GROUPS

Abstract

The real Torelli mapping, from the moduli space of real curves of genus g to the moduli space of g-dimensional real principally polarized abelian varieties, sends a real curve into its real Jacobian. The real Schottky problem is to describe its image. The results contained in the present paper concern hyperelliptic real curves and in particular real curves of genus 2. We exhibit also some counterexamples for the non-hyperelliptic case.     

Numerical Schottky Uniformizations

A real algebraic curve of algebraic genus g ≥ 2 is a pair (S, τ), where S is a closed Riemann surface of genus g and τ is a reflection on S (anticonformal involution with fixed points). In this note, we discuss a numerical (Burnside) program which permits to obtain a Riemann period matrix of the surface S in terms of an uniformizing real Schottky group. If we denote by Aut₊(S, τ) the group of conformal automorphisms of S commuting with the real structure τ, then it is a well known fact that |Aut₊(S,τ)| ≥ 12(g−1). We say that (S,τ) is maximally symmetric if |Aut₊(S,τ)|=12(g−1). We work explicitly such a numerical program in the case of maximally symmetric real curves of genus two. We construct a real Schottky uniformization for each such real curve and we use the numerical program to obtain a real algebraic curve, a Riemann period matrix and the accessory parameters in terms of the corresponding Schottky uniformization. In particular, we are able to give for Bolza’s curve a Schottky uniformization (at least numerically), providing an example for which the inverse uniformization theorem is numerically solved.Partially supported by Projects Fondecyt 1030252 1030373 and UTFSM 12.03.21     

Totally real embeddings of the torus intoC 2

In this short note we combine a construction of Viro and a result of Eliashberg and Harlamov to prove that there exist smooth totally real embeddings of the torus intoC ² which are isotopic but not so within the class of totally real surfaces. We also show how Viro's construction can be used to define an isotopy invariant for a certain class of complex curves inC P ².     

Edge Matrix of Hyperbolic Simplices

In this paper, by using the dual problem which was solved by Feng Luo (Geom. Dedicata 64 (1997), 277–282) and a new method, we give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a hyperbolic n-simplex. By using determinants, we also give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a spherical n-simplex.Mathematics Subject Classifications (2000). 51M04, 51M05, 51M20, 51M25, 52A38, 52A37, 52B10.     

On $$$$-Hypermonogenic Functions and Their Mean Value Properties

We study a hyperbolic version of holomorphic functions to higher dimensions. In this frame work, a generalization of holomorphic functions are called \(k\)-hypermonogenic functions. These functions are depending on several real variables and their values are in a Clifford algebra. They are defined in terms of hyperbolic Dirac operators. They are connected to harmonic functions with respect to the Riemannian metric \(ds^{2}_k=x_{n}^{2k/\left( 1-n\right) }\left( dx_{0}^{2}+\cdots +dx_{n}^{2}\right) \) in the same way as the usual harmonic function to holomorphic functions. We present the mean value property for \(k\)-hypermonogenic functions and related results. Earlier the mean value properties has been proved for hypermonogenic functions. The key tools are the invariance properties of the hyperbolic metric.     

The canonical decomposition of once-punctured torus bundles

We determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defined by Floyd and Hatcher. Unlike previous work on this problem, the techniques we use are not geometric. Instead, they involve angled polyhedral decompositions, thin position and a version of almost normal surface theory.     

Some algebraic properties of the hyperbolic systems

Abstract The technique of quasi-symmetrizer has been applied to the well-posedness of the Cauchy problem for scalar operators [10], [13] and linear systems [5], [15], [4], and to the propagation of analitycity for solutions to semi-linear systems [6]. In all these works, it is assumed that the principal symbol depends only on the time variable. In this note we illustrate, in some special cases, a new property of the quasisymmetrizer which allows us to generalize the result in [6] to semi-linear systems with coefficients depending also on the space variables [21]. Such a property is closely connected with some interesting inequalities on the eigenvalues of a hyperbolic matrix. We expect that this technique applies also to other problems. Keywords: First order hyperbolic systems, Quasi-symmetrizer, Glaeser inequality     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

Singular sets of planar hyperbolic billiards are regular

Many planar hyperbolic billiards are conjectured to be ergodic. This paper represents a first step towards the proof of this conjecture. The Hopf argument is a standard technique for proving the ergodicity of a smooth hyperbolic system. Under additional hypotheses, this technique also applies to certain hyperbolic systems with singularities, including hyperbolic billiards. The supplementary hypotheses concern the subset of the phase space where the system fails to be C ² differentiable. In this work, we give a detailed proof of one of these hypotheses for a large collection of planar hyperbolic billiards. Namely, we prove that the singular set and each of its iterations consist of a finite number of compact curves of class C ² with finitely many intersection points.     

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.     

Inversion de la transformation de Pompeiu locale dans les espaces hyperboliques reel et complexe: Cas des deux boules

In this paper we extend the result we have established for the hyperbolic disk in [8] to the real and complex hyperbolic spaces. This includes the reconstruction of a function defined in a fixed ball from its averages on balls of radiir ₁,r ₂ lying inB(0,R).      

A Note on Purely Imaginary Roots of Generalized Kac–Moody Algebras

Abstract

In this paper, we generalize the concept of purely imaginary roots of Kac–Moody algebras to generalized Kac–Moody algebras. Also we give a complete classification of those generalized Kac–Moody algebras with the purely imaginary property. We also define a new class of indefinite non-hyperbolic generalized Kac–Moody algebras called extended hyperbolic generalized Kac–Moody algebras and find that it does not always possess the purely imaginary property whereas the extended hyperbolic Kac–Moody algebras possess the purely imaginary property.     

On pseudocyclic table algebras and applications to pseudocyclic association schemes

We effect a complete study of the thermodynamic formalism, the entropy spectrum of Birkhoff averages, and the ergodic optimization problem for a family of parabolic horseshoes. We consider a large class of potentials that are not necessarily regular, and we describe both the uniqueness of equilibrium measures and the occurrence of phase transitions for nonregular potentials in this class. Our approach consists in reducing the problems to the study of renewal shifts. We also describe applications of this approach to hyperbolic horseshoes as well as to noninvertible maps, both parabolic (with the Manneville-Pomeau map) and uniformly expanding. This allows us to recover in a unified manner several results scattered in the literature. For the family of hyperbolic horseshoes, we also describe the dimension spectrum of equilibrium measures of a class of potentials that are not necessarily regular. In particular, the dimension spectra need not be strictly convex.     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.