Pattern rigidity in hyperbolic spaces: duality and PD subgroups

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.