Geometry & Topology, Vol. 8 (2004) Paper no. 5, pages 205--275.

Nonpositively curved 2-complexes with isolated flats

G Christopher Hruska

Abstract. We introduce the class of nonpositively curved 2-complexes with the Isolated Flats Property. These 2-complexes are, in a sense, hyperbolic relative to their flats. More precisely, we show that several important properties of Gromov-hyperbolic spaces hold `relative to flats' in nonpositively curved 2-complexes with the Isolated Flats Property.
We introduce the Relatively Thin Triangle Property, which states roughly that the fat part of a geodesic triangle lies near a single flat. We also introduce the Relative Fellow Traveller Property, which states that pairs of quasigeodesics with common endpoints fellow travel relative to flats, in a suitable sense. The main result of this paper states that in the setting of CAT(0) 2-complexes, the Isolated Flats Property is equivalent to the Relatively Thin Triangle Property and is also equivalent to the Relative Fellow Traveller Property.

Keywords. Word hyperbolic, nonpositive curvature, thin triangles, quasigeodesics, isolated flats

AMS subject classification. Primary: 20F67. Secondary: 20F06, 57M20.

DOI: 10.2140/gt.2004.8.205

E-print: arXiv:math.MG/0402231

Submitted to GT on 22 January 2003. (Revised 12 February 2004.) Paper accepted 17 December 2003. Paper published 12 February 2004.

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G Christopher Hruska
Department of Mathematics, University of Chicago
5734 S University Ave, Chicago, IL 60637, USA

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