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.   We consider discrete cocompact isometric actions  where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action  which determine, and are determined by, the equivariant homeomorphism type of the action  of G on the ideal boundary of X. Moreover, if  are two actions with the same geometric data and  is a G-equivariant quasi-isometry, then for every geodesic ray  there is a geodesic ray  (unique up to equivalence) so that . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136].

The geodesic flow of a nonpositively curved graph manifold