Relative Rigidity, Quasiconvexity and C-Complexes

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.Comment: 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

Height in splittings of hyperbolic groups

Suppose $H$ is a hyperbolic subgroup of a hyperbolic group $G$. Assume there exists $n > 0$ such that the intersection of $n$ essentially distinct conjugates of $H$ is always finite. Further assume $G$ splits over $H$ with hyperbolic vertex and edge groups and the two inclusions of $H$ are quasi-isometric embeddings. Then $H$ is quasiconvex in $G$. This answers a question of Swarup and provides a partial converse to the main theorem of \cite{GMRS}.Comment: 16 pages, no figures, no table