The geodesic flow of a nonpositively curved graph manifold

Abstract.

We consider discrete cocompact isometric actions $G \curvearrowright^{\rho} X$ 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 $\rho$ which determine, and are determined by, the equivariant homeomorphism type of the action $G \curvearrowright^{\partial_\infty \rho}\,\partial_\infty X X$ of G on the ideal boundary of X. Moreover, if $G \curvearrowright^{\rho}i X_i$ are two actions with the same geometric data and $\Phi : X_1 \to X_2$ is a G-equivariant quasi-isometry, then for every geodesic ray $\gamma_1 : [0, \infty) \to X_1$ there is a geodesic ray $\gamma_2 : [0, \infty) \to X_2$ (unique up to equivalence) so that ${\rm lim}_{t \to \infty} {1 \over t}\,d_{X_2}(\Phi \circ \gamma_1(t), \gamma_2([0, \infty))) = 0$. This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136].