On the horoboundary and the geometry of rays of negatively curved manifolds

We study the Gromov compactification of quotients X=G of a Hadamard space X by a discrete group of isometries G, pointing out the main differences with the simply connected case. We prove a criterion for the Busemann equivalence of rays on these quotients and show that the "visual" description of the Gromov boundary breaks down, producing examples for the main pathologies that may occur in the nonsimply connected case, such as: divergent rays having the same Busemann functions, points on the Gromov boundary that are not Busemann functions of any ray, and discontinuity of the Busemann functions with respect to the initial conditions. Finally, for geometrically finite quotients X=G, we recover a simple description of the Gromov boundary, and prove that in this case the compactification is a singular manifold with boundary, with a finite number of conical singularities. © 2012 by Pacific Journal of Mathematics.