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].