On the Gromov compactification and the geometry of rays of (non-simply connected) negatively curved manifolds.

The notion of Busemann function was originally introduced by Herbert Busemann in the fifties as a tool to develop a theory of parallels on geodesic spaces (e.g. complete Riemannian manifolds). The Busemann functions captured the idea of angle at infinity between infinite geodesic rays, and this idea played an important role in the study of the
topology complete noncompact Riemannian manifolds.
In particular this notion has a special place in the geometry of Hadamard spaces (simply connected manifolds with nonpositive curvature) and in the dynamics of Kleinian groups. For a Hadamard manifold X, the Busemann
functions yield a useful compactification of the space, which was originally introduced by M.Gromov; this compactification has the topology of a sphere and is easily understood in terms of rays. This nice picture breaks down for non-simply connected manifolds. The aim of the talk is to explain the visual description of the Gromov compactification and the Gromov boundary for Hadamard spaces and the main differences with the non-simply connected case.

We will interpret the Buseman equivalence on a quotient of a Hadamard manifold G\X in terms of the action of G on the universal covering, and we will give some examples of the main interesting pathologies in the non-simply connected case, namely:

-- divergent rays having the same Busemann functions;

-- points on the Gromov boundary which are not Busemann functions of any ray;

-- discontinuity of the Busemann functions with respect to the initial conditions.