Relative rigidity, quasiconvexity and C-complexes

Abstract

We introduce and study the notion of relative rigidity for pairs (X, J ) where 1) X is a hyperbolic metric space and J a collection of quasiconvex sets 2) X is a relatively hyperbolic group and J the collection of parabolics 3) X is a higher rank symmetric space and J an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such J'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, J ) 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.