#### Volume 8, issue 3 (2008)

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Relative rigidity, quasiconvexity and $C$–complexes

### Mahan Mj

Algebraic & Geometric Topology 8 (2008) 1691–1716
##### Abstract

We introduce and study the notion of relative rigidity for pairs $\left(X,\mathsc{J}\right)$ where

(1)  $X$ is a hyperbolic metric space and $\mathsc{J}$ a collection of quasiconvex sets,

(2)  $X$ is a relatively hyperbolic group and $\mathsc{J}$ the collection of parabolics,

(3)  $X$ is a higher rank symmetric space and $\mathsc{J}$ an equivariant collection of maximal flats.

Relative rigidity can roughly be described as upgrading a uniformly proper map between two such $\mathsc{J}$ to a quasi-isometry between the corresponding $X$. 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 $\left(X,\mathsc{J}\right)$ 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.

##### Keywords
Hyperbolic group, Quasiconvex subgroup, flats, relative hyperbolicity
##### Mathematical Subject Classification 2000
Primary: 20F67
Secondary: 57M50, 22E40