#### Volume 8, issue 1 (2004)

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Nonpositively curved 2–complexes with isolated flats

### G Christopher Hruska

Geometry & Topology 8 (2004) 205–275
 arXiv: math.MG/0402231
##### 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 $CAT\left(0\right)$ 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
##### Mathematical Subject Classification 2000
Primary: 20F67
Secondary: 20F06, 57M20
##### Publication
Received: 22 January 2003
Revised: 12 February 2004
Accepted: 17 December 2003
Published: 12 February 2004
Proposed: Benson Farb
Seconded: Walter Neumann, Martin Bridson
##### Authors
 G Christopher Hruska Department of Mathematics University of Chicago 5734 South University Ave Chicago Illinois 60637 USA