ar X iv : d g - ga / 9 40 70 01 v 1 1 J ul 1 99 4 TEICHMÜLLER SPACE IS NOT GROMOV HYPERBOLIC
نویسنده
چکیده
The Teichmüller space of surfaces of genus g > 1 with the Teichmüller metric is not nonpositively curved, in the sense that there are distinct geodesic rays from a point that always remain within a bounded distance of each other ([Ma1].) Despite this phenomenon, Teichmüller space and its quotient, Moduli space, share many properties with spaces of negative curvature: for instance, most converging geodesic rays are asymptotic [Ma2], and the geodesic flow on the moduli space is ergodic [Ma3]. One can ask whether these properties can be explained by Teichmüller space having non-positive curvature in a sense weaker than that of Busemann used in [Ma1], which declared a space X to be negatively curved if the endpoints of two segments from p ∈ X are spread more than twice as far as the midpoints. In this study of hyperbolic groups, Gromov ([Gr], see also [GdlH]) introduced a notion of negative curvature, now called Gromov hyperbolicity, that still captured many of the qualitative aspects of Riemannian negative sectional curvature, but was less restrictive than that of Busemann. Specifically, Gromov declared a space X to be hyperbolic if there existed a number M so that for any p ∈ X and any triangle in X with vertex at p, the leg of the triangle opposite p would be within an M -neighborhood of the legs of the triangle emanating from p. Thus, for instance, the flat Euclidean strip {(x, y) ∈ R | 0 < x < 1} would be Gromov hyperbolic but not Buseman negatively curved; moreover, the fact that there are pairs of rays emanating from p ∈ Tg which do not diverge does not, in itself, preclude Teichmüller space with the Teichmüller metric from being Gromov hyperbolic.
منابع مشابه
ar X iv : g r - qc / 9 40 70 01 v 1 4 J ul 1 99 4 Cat States and Single Runs for the Damped Harmonic Oscillator
We discuss the fate of initial states of the cat type for the damped harmonic oscillator, mostly employing a linear version of the stochastic Schrödinger equation. We also comment on how such cat states might be prepared and on the relation of single realizations of the noise to single runs of experiments. Typeset using REVTEX 1
متن کاملar X iv : d g - ga / 9 70 70 17 v 1 2 4 Ju l 1 99 7 SCALAR CURVATURE RIGIDITY FOR ASYMPTOTICALLY LOCALLY HYPERBOLIC MANIFOLDS
Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expressi...
متن کاملar X iv : m at h / 04 07 08 6 v 1 [ m at h . G T ] 6 J ul 2 00 4 REPRESENTATIONS OF THE QUANTUM TEICHMÜLLER SPACE AND INVARIANTS OF SURFACE DIFFEOMORPHISMS
— We investigate the representation theory of the polynomial core T S of the quantum Teichmüller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of T S are classified, up to finitely many choices, by group hom...
متن کاملar X iv : d g - ga / 9 60 30 13 v 1 2 4 M ar 1 99 6 Harmonic manifolds with some specific volume densities
We show that noncompact simply connected harmonic manifolds with volume density Θ p (r) = sinh n−1 r is isometric to the real hyperbolic space and noncompact simply connected Kähler harmonic manifold with volume density Θ p (r) = sinh 2n−1 r cosh r is isometric to the complex hyperbolic space. A similar result is also proved for Quaternionic Kähler manifolds. Using our methods we get an alterna...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1995