Nonpositively curved 2-complexes with isolated flats

Details Nonpositively curved 2-complexes with isolated flats Nonpositively curved 2-complexes with isolated flats

2004-02-13

arxiv.org/abs/math/0402231v1

Geom. Topol. 8(2004) 205-275

Mathematics

Metric Geometry

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper5.abs.html

Scientific paper

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.

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