### Abstract

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: X_{1} → X_{2} is a G-equivariant quasi-isometry, then for every geodesic ray γ_{1} : [0, ∞) → X_{1}, there is a geodesic ray γ_{2} : [0, ∞) → X_{2} (unique up to equivalence) so that lim_{t→∞} 1/td_{X2} (Φ O γ_{1}(t), γ_{2}([0, ∞))) = 0. This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136].

Original language | English (US) |
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Pages (from-to) | 479-545 |

Number of pages | 67 |

Journal | Geometric and Functional Analysis |

Volume | 12 |

Issue number | 3 |

DOIs | |

State | Published - Dec 3 2002 |

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology

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## Cite this

*Geometric and Functional Analysis*,

*12*(3), 479-545. https://doi.org/10.1007/s00039-002-8255-7