We introduce and study the notion of relative rigidity for pairs where
(1) is a hyperbolic metric space and a collection of quasiconvex sets,
(2) is a relatively hyperbolic group and the collection of parabolics,
(3) is a higher rank symmetric space and an equivariant collection of maximal flats.
Relative rigidity can roughly be described as upgrading a uniformly proper map between two such to a quasi-isometry between the corresponding . A related notion is that of a –complex which is the adaptation of a Tits complex to this context. We prove the relative rigidity of the collection of pairs 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 –complexes. We also give a couple of characterizations of quasiconvexity of subgroups of hyperbolic groups on the way.
"Relative rigidity, quasiconvexity and $C$–complexes." Algebr. Geom. Topol. 8 (3) 1691 - 1716, 2008. https://doi.org/10.2140/agt.2008.8.1691