Systole et rayon maximal des vari\'et\'es hyperboliques non compactes

TL;DR

This paper establishes bounds on the systole and inradius of cusped hyperbolic manifolds, providing sharp results in dimension 3 and new characterizations of specific manifolds.

Contribution

It introduces new bounds for the systole and inradius of cusped hyperbolic manifolds, including sharp bounds in dimension 3 and a novel characterization of the Gieseking manifold.

Findings

Upper bound for systole in terms of dimension and simplicial volume

Positive lower bound on inradius independent of dimension

Sharp bounds in dimension 3 realized by the Gieseking manifold

Abstract

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 3, realized by the Gieseking manifold. It provides a new characterization of this manifold.