Nonpositively curved 2–complexes with isolated flats

Hruska, G Christopher¹

¹ Department of Mathematics, University of Chicago, 5734 South University Ave, Chicago, Illinois 60637, USA

Geometry & topology, Tome 8 (2004) no. 1, pp. 205-275.

Résumé

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.

Détail

DOI : 10.2140/gt.2004.8.205

Keywords: word hyperbolic, nonpositive curvature, thin triangles, quasigeodesics, isolated flats

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