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
Email: chruska@math.uchicago.edu, bkleiner@umich.edu, mhindawi@math.upenn.edu
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