Geometry & Topology

Nonpositively curved 2–complexes with isolated flats

G Christopher Hruska

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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.

Article information

Geom. Topol., Volume 8, Number 1 (2004), 205-275.

Received: 22 January 2003
Revised: 12 February 2004
Accepted: 17 December 2003
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F06: Cancellation theory; application of van Kampen diagrams [See also 57M05] 57M20: Two-dimensional complexes

word hyperbolic nonpositive curvature thin triangles quasigeodesics isolated flats


Hruska, G Christopher. Nonpositively curved 2–complexes with isolated flats. Geom. Topol. 8 (2004), no. 1, 205--275. doi:10.2140/gt.2004.8.205.

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