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

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.

Keywords

isolated flats, asymptotic cone, relative hyperbolicity

Mathematical Subject Classification 2000

Primary: 20F67

Secondary: 20F69

References

Forward citations

Publication

Received: 5 April 2005

Revised: 25 July 2005

Accepted: 24 June 2005

Published: 8 August 2005

Proposed: Walter Neumann

Seconded: Martin Bridson, Benson Farb

Correction: 1 January 2009

Authors

G Christopher Hruska
Department of Mathematics University of Chicago 5734 S University Ave Chicago Illinois 60637-1514 USA

Bruce Kleiner
Department of Mathematics University of Michigan Ann Arbor Michigan 48109-1109 USA

Volume 9, issue 3 (2005)

G Christopher Hruska and Bruce Kleiner

Abstract