TY - JOUR

T1 - The geodesic flow of a nonpositively curved graph manifold

AU - Croke, C. B.

AU - Kleiner, B.

N1 - Funding Information:
C.B.C. supported by NSF grants DMS-95-05175 DMS-96-26-232 and DMS 99-71749. B.K. supported by a Sloan Foundation Fellowship, and NSF grants DMS-95-05175, DMS-96-26911, DMS-9022140.

PY - 2002

Y1 - 2002

N2 - 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].

AB - 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].

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U2 - 10.1007/s00039-002-8255-7

DO - 10.1007/s00039-002-8255-7

M3 - Article

AN - SCOPUS:0035995734

SN - 1016-443X

VL - 12

SP - 479

EP - 545

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 3

ER -