Relative Rigidity, Quasiconvexity and C-Complexes

Abstract

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.

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"

References in corpus (3)

• Hadamard spaces with isolated flats
• Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
• Quasi-actions on trees II: Finite depth Bass-Serre trees

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