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For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space$\Hyp^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infiniteindex quasiconvex subgroups satisfying one of the following conditions: 1)limit set of $H_i$ is a codimension one topological sphere. 2) limit set of$H_i$ is an even dimensional topological sphere. 3) $H_i$ is a codimension oneduality group. This generalizes (1). In particular, if $n = 3$, $H_i$ could beany freely indecomposable subgroup of $G_i$. 4) $H_i$ is an odd-dimensionalPoincare Duality group $PD(2k+1)$. This generalizes (2). We prove patternrigidity for such pairs extending work of Schwartz who proved pattern rigiditywhen $H_i$ is cyclic. All this generalizes to quasiconvex subgroups of uniformlattices in rank one symmetric spaces satisfying one of the conditions (1)-(4),as well as certain special subgroups with disconnected limit sets. Inparticular, pattern rigidity holds for all quasiconvex subgroups of hyperbolic3-manifolds that are not virtually free. Combining this with a result ofMosher-Sageev-Whyte, we get quasi-isometric rigidity results for graphs ofgroups where the vertex groups are uniform lattices in rank one symmetricspaces and edge groups are of any of the above types.

Pattern Rigidity in Hyperbolic Spaces: Duality and PD Subgroups