Geometry & Topology, Vol. 9 (2005) Paper no. 34, pages 1501--1538.

Hadamard spaces with isolated flats

G Christopher Hruska and Bruce Kleiner
with an appendix written jointly with Mohamad Hindawi

Abstract. 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

AMS subject classification. Primary: 20F67. Secondary: 20F69.

E-print: arXiv:math.GR/0411232

DOI: 10.2140/gt.2005.9.1501

Submitted to GT on 5 April 2005. (Revised 25 July 2005.) Paper accepted 24 June 2005. Paper published 8 August 2005.

Notes on file formats

G Christopher Hruska, Bruce Kleiner, Mohamad Hindawi

GCH: Department of Mathematics, University of Chicago
5734 S University Ave, Chicago, IL 60637-1514, USA

BK: Department of Mathematics, University of Michigan
Ann Arbor, MI 48109-1109, USA

MH: Department of Mathematics, University of Pennsylvania
Philadelphia, PA 19104-6395, USA


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