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Hadamard spaces with isolated flats, with an appendix written jointly with Mohamad Hindawi

G Christopher Hruska and Bruce Kleiner

Geometry & Topology 9 (2005) 1501–1538

Erratum: Geometry & Topology 13 (2009) 699–707

arXiv: math.GR/0411232


We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space is an invariant of the group up to equivariant homeomorphism. We also prove that any such group is relatively hyperbolic, biautomatic, and satisfies the Tits Alternative. The main step in establishing these results is a characterization of spaces with isolated flats as relatively hyperbolic with respect to flats. Finally we show that a CAT(0) space has isolated flats if and only if its Tits boundary is a disjoint union of isolated points and standard Euclidean spheres. In an appendix written jointly with Hindawi, we extend many of the results of this article to a more general setting in which the isolated subspaces are not required to be flats.

isolated flats, asymptotic cone, relative hyperbolicity
Mathematical Subject Classification 2000
Primary: 20F67
Secondary: 20F69
Forward citations
Received: 5 April 2005
Revised: 25 July 2005
Accepted: 24 June 2005
Published: 8 August 2005
Proposed: Walter Neumann
Seconded: Martin Bridson, Benson Farb
G Christopher Hruska
Department of Mathematics
University of Chicago
5734 S University Ave
Illinois 60637-1514
Bruce Kleiner
Department of Mathematics
University of Michigan
Ann Arbor
Michigan 48109-1109