Pattern rigidity in hyperbolic spaces: duality and PD subgroups

Kingshook, Biswas ; Mahan, MJ (2012) Pattern rigidity in hyperbolic spaces: duality and PD subgroups Groups, Geometry, and Dynamics, 6 (1). pp. 97-123. ISSN 1661-7207

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Abstract

For i = 1, 2, let Gi be cocompact groups of isometries of hyperbolic space Hn of real dimension n, n ≥ 3. Let Hi ⊂ Gi be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of Hi is a codimension one topological sphere. 2) limit set of Hi is an even dimensional topological sphere. 3) Hi is a codimension one duality group. This generalizes (1). In particular, if n = 3, Hi could be any freely indecomposable subgroup of Gi. 4) Hi is an odd-dimensional Poincare Duality group PD(2k + 1). This gener alizes (2). We prove pattern rigidity for such pairs extending work of Schwartz who proved pattern rigidity when Hi is cyclic. All this generalizes to quasiconvex subgroups of uniform lattices in rank one symmetric spaces satisfying one of the conditions (1)-(4), as well as certain special subgroups with disconnected limit sets. In particular, pattern rigidity holds for all quasiconvex subgroups of hyperbolic 3-manifolds that are not virtually free. Combining this with the main result of Mosher-Sageev-Whyte [MSW04], we get quasi-isometric rigidity results for graphs of groups where the vertex groups are uniform lattices in rank one symmetric spaces and edge groups are of any of the above types.

Item Type:Article
Source:Copyright of this article belongs to European Mathematical Society.
ID Code:89536
Deposited On:28 Apr 2012 12:57
Last Modified:19 May 2016 04:03

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