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

Cannon-Thurston Maps,i-bounded Geometry and a theorem of McMullen

The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map. This is an exposition of a special case of the main result of arXiv:math/0607509.Comment: v3: 32 pages 3 figure