Nonpositively curved 2–complexes with isolated flats

Hruska, G Christopher

doi:10.2140/gt.2004.8.205

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Nonpositively curved 2–complexes with isolated flats

G Christopher Hruska

Geometry & Topology 8 (2004) 205–275

DOI: 10.2140/gt.2004.8.205

arXiv: math.MG/0402231

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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 20F67

Secondary: 20F06, 57M20

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Publication

Received: 22 January 2003

Revised: 12 February 2004

Accepted: 17 December 2003

Published: 12 February 2004

Proposed: Benson Farb

Seconded: Walter Neumann, Martin Bridson

Authors

G Christopher Hruska
Department of Mathematics University of Chicago 5734 South University Ave Chicago Illinois 60637 USA

Volume 8, issue 1 (2004)