Systole et rayon interne des variétés hyperboliques non compactes

Gendulphe, Matthieu¹

¹ Département de Mathématiques, Université de Fribourg, Chemin du Musée 23, CH-1700 Fribourg Pérolles, Switzerland

Geometry & topology, Tome 19 (2015) no. 4, pp. 2039-2080.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.2039

Classification : 57M50, 30F45
Keywords: hyperbolic manifolds, cusps, systole, inradius, injectivity radius

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