Coarse-geometric perspective on negatively curved manifolds and groups

Let (X,g) be a compact negatively curved Riemannian manifold with fundamental group G. Restricting the lifted metric on the universal cover Y of X to a G-orbit Gx one obtains a left invariant metric on G, which is well defined up to a bounded amount, depending on the choice of the orbit Gx. Motivated by this geometric example, we study classes [d] of general left-invariant metrics d on general Gromov hyperbolic groups G, 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, cross-ratio, 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 the coarse-geometric setting.

Authors: A. Furman,Bibliographical: in

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[Papers] [Alex Furman]