Relative Rigidity, Quasiconvexity and C-Complexes

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