Relative Rigidity, Quasiconvexity and C-Complexes

Details Relative Rigidity, Quasiconvexity and C-Complexes Relative Rigidity, Quasiconvexity and C-Complexes

2007-04-16

arxiv.org/abs/0704.1922v4

Algebraic & Geometric Topology 8 (2008) 1691-1716

Mathematics

Geometric Topology

23pgs, v3: Relative rigidity proved for relatively hyperbolic groups and higher rank symmetric spaces, v4: final version incor

Scientific paper

10.2140/agt.2008.8.1691

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.

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