# Groups, Geometry, and Dynamics

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**Volume 6, Issue 1, 2012, pp. 97–123**

**DOI: 10.4171/GGD/152**

Published online: 2012-02-08

Pattern rigidity in hyperbolic spaces: duality and PD subgroups

Kingshook Biswas^{[1]}and Mahan Mj

^{[2]}(1) RKM Vivekananda University, Dist. Howrah, West Bengal, India

(2) Tata Institute of Fundamental Research, Mumbai, India

For $i= 1,2$, let $G_i$ be cocompact groups
of isometries of hyperbolic space
$\mathbf{H}^n$ of real dimension $n$, $n \geq 3$.
Let $H_i \subset G_i$ be
infinite index quasiconvex subgroups satisfying one of the following
conditions:

- The limit set of $H_i$ is a codimension one topological sphere.
- The limit set of $H_i$ is an even dimensional topological sphere.
- $H_i$ is a codimension one
*duality group*. This generalizes (1). In particular, if $n = 3$, $H_i$ could be*any*freely indecomposable subgroup of $G_i$. - $H_i$ is an odd-dimensional PoincarĂ© duality group PD$(2k+1)$. This generalizes (2).

We prove pattern rigidity for such pairs extending work of Schwartz who proved pattern rigidity when $H_i$ is cyclic. All this generalizes to quasiconvex subgroups of uniform lattices in rank one symmetric spaces satisfying one of the conditions (1)â€“(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 results of Mosher, Sageev, and Whyte, we obtain quasi-isometric rigidity results for graphs of groups where the vertex groups are uniform lattices in rank one symmetric spaces and the edge groups are of any of the above types.

*Keywords: *Quasi-isometric rigidity, pattern rigidity, homology manifold, quasiconformal map

Biswas Kingshook, Mj Mahan: Pattern rigidity in hyperbolic spaces: duality and PD subgroups. *Groups Geom. Dyn.* 6 (2012), 97-123. doi: 10.4171/GGD/152