## Geometry & Topology

### Nonpositively curved 2–complexes with isolated flats

G Christopher Hruska

#### 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 $CAT(0)$ 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

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

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

https://projecteuclid.org/euclid.gt/1513883366

Digital Object Identifier
doi:10.2140/gt.2004.8.205

Mathematical Reviews number (MathSciNet)
MR2033482

Zentralblatt MATH identifier
1063.20048

#### Citation

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. https://projecteuclid.org/euclid.gt/1513883366

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