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2008 Relative rigidity, quasiconvexity and $C$–complexes
Mahan Mj
Algebr. Geom. Topol. 8(3): 1691-1716 (2008). DOI: 10.2140/agt.2008.8.1691

Abstract

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.

Citation

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Mahan Mj. "Relative rigidity, quasiconvexity and $C$–complexes." Algebr. Geom. Topol. 8 (3) 1691 - 1716, 2008. https://doi.org/10.2140/agt.2008.8.1691

Information

Received: 16 August 2007; Revised: 1 August 2008; Accepted: 3 August 2008; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1179.20039
MathSciNet: MR2448868
Digital Object Identifier: 10.2140/agt.2008.8.1691

Subjects:
Primary: 20F67
Secondary: 22E40, 57M50

Rights: Copyright © 2008 Mathematical Sciences Publishers

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