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Relative rigidity, quasiconvexity and $C$–complexes

Mahan Mj

Algebraic & Geometric Topology 8 (2008) 1691–1716

We introduce and study the notion of relative rigidity for pairs (X,J ) where

(1)  X is a hyperbolic metric space and J a collection of quasiconvex sets,

(2)  X is a relatively hyperbolic group and J the collection of parabolics,

(3)  X is a higher rank symmetric space and J an equivariant collection of maximal flats.

Relative rigidity can roughly be described as upgrading a uniformly proper map between two such 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 (X,J ) 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.

Hyperbolic group, Quasiconvex subgroup, flats, relative hyperbolicity
Mathematical Subject Classification 2000
Primary: 20F67
Secondary: 57M50, 22E40
Received: 16 August 2007
Revised: 1 August 2008
Accepted: 3 August 2008
Published: 8 October 2008
Mahan Mj
School of Mathematical Sciences
RKM Vivekananda University
PO Belur Math
Dt Howrah