The geodesic flow of a nonpositively curved graph manifold

We consider discrete cocompact isometric actions G right curved arrow sign^ρ X where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces - complete, simply connected length spaces with non-positive 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 p which determine, and are determined by, the equivariant homeomorphism type of the action G right curved arrow sign^∂∞ρ ∂_∞X of G on the ideal boundary of X. Moreover, if G right curved arrow sign^ρi X_i are two actions with the same geometric data and φ: X₁ → X₂ is a G-equivariant quasi-isometry, then for every geodesic ray γ₁ : [0, ∞) → X₁, there is a geodesic ray γ₂ : [0, ∞) → X₂ (unique up to equivalence) so that lim_t→∞ 1/td_X2 (Φ O γ₁(t), γ₂([0, ∞))) = 0. This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136].