We consider discrete cocompact isometric actions G right curved arrow signρ 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 signρi Xi 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 limt→∞ 1/tdX2 (Φ O γ1(t), γ2([0, ∞))) = 0. This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136].
ASJC Scopus subject areas
- Geometry and Topology