Volume 19, issue 4 (2015)

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Systole et rayon interne des variétés hyperboliques non compactes

Matthieu Gendulphe

Geometry & Topology 19 (2015) 2039–2080
Abstract

Nous contrôlons deux invariants globaux des variétés hyperboliques à bouts cuspidaux : la longueur de la plus courte géodésique fermée (la systole), et le rayon de la plus grande boule plongée (le rayon maximal). Nous majorons la systole en fonction de la dimension et du volume simplicial. Nous minorons le rayon maximal par une constante positive indépendante de la dimension. Ces bornes sont optimales en dimension $\mathfrak{3}$. Cela donne une nouvelle caractérisation de la variété de Gieseking.

We bound two global invariants of cusped hyperbolic manifolds: the length of the shortest closed geodesic (the systole), and the radius of the biggest embedded ball (the inradius). We give an upper bound for the systole, expressed in terms of the dimension and simplicial volume. We find a positive lower bound on the inradius independent of the dimension. These bounds are sharp in dimension $\mathfrak{3}$, realized by the Gieseking manifold. They provide a new characterization of this manifold.

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Keywords
hyperbolic manifolds, cusps, systole, inradius, injectivity radius
Primary: 57M50
Secondary: 30F45
Publication
Received: 21 June 2013
Revised: 11 September 2014
Accepted: 13 October 2014
Published: 29 July 2015
Proposed: Jean-Pierre Otal
Seconded: Leonid Polterovich, Martin R Bridson
Authors
 Matthieu Gendulphe Département de Mathématiques Université de Fribourg Chemin du Musée 23 CH-1700 Fribourg Pérolles Switzerland http://Matthieu.Gendulphe.com