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Nonpositively curved 2–complexes with isolated flats

G Christopher Hruska

Geometry & Topology 8 (2004) 205–275

arXiv: math.MG/0402231


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.

word hyperbolic, nonpositive curvature, thin triangles, quasigeodesics, isolated flats
Mathematical Subject Classification 2000
Primary: 20F67
Secondary: 20F06, 57M20
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Received: 22 January 2003
Revised: 12 February 2004
Accepted: 17 December 2003
Published: 12 February 2004
Proposed: Benson Farb
Seconded: Walter Neumann, Martin Bridson
G Christopher Hruska
Department of Mathematics
University of Chicago
5734 South University Ave
Illinois 60637