We introduce and study the notion of relative rigidity for pairs (X,\JJ)
where 1) X is a hyperbolic metric space and \JJ a collection of quasiconvex
sets 2) X is a relatively hyperbolic group and \JJ the collection of
parabolics 3) X is a higher rank symmetric space and \JJ an equivariant
collection of maximal flats Relative rigidity can roughly be described as
upgrading a uniformly proper map between two such \JJ's to a quasi-isometry
between the corresponding X's.
A related notion is that of a C-complex which is the adaptation of a Tits
complex to this context. We prove the relative rigidity of the collection of
pairs (X, \JJ) as above. This generalises a result of Schwarz for symmetric
patterns of geodesics in hyperbolic space. We show that a uniformly proper map
induces an isomorphism of the corresponding C-complexes. We also give a
couple of characterizations of quasiconvexity. of subgroups of hyperbolic
groups on the way.Comment: 23pgs, v3: Relative rigidity proved for relatively hyperbolic groups
and higher rank symmetric spaces, v4: final version incorporating referee's
comments. To appear in "Algebraic and Geometric Topology
