Coarse-Geometric Perspective on Negatively Curved Manifolds and Groups

Abstract

Let (X, g) be a compact negatively curved Riemannian manifold with fundamental group Γ. Restricting the lifted metric on the universal cover $\left( {\tilde X,\tilde g} \right)$ of (X, g) to a Γ-orbit, Γx, one obtains a left invariant metric d _{g, x} on Γ, which is well defined up to a bounded amount, depending on the choice of the orbit Γx. Motivated by this geometric example, we study classes [d]of general left-invariant metrics d on general Gromov hyperbolic groups Γ, where [d ₁]= [d ₂]if d ₁ — d ₂ is bounded. It turns out that many of the geometric objects associated with (X, g) — such as marked length spectrum, crossratio, Bowen-Margulis measure — can be defined in the general coarse-geometric setting. The main result of the paper is a characterization of the compact negatively curved locally symmetric spaces within this coarse-geometric setting.

The author was partially supported by NSF grants DMS-9803607, 0049069 and CNRS.