  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.
