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
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
