Pattern rigidity in hyperbolic spaces: duality and PD subgroups

Abstract

For i = 1, 2, let G_i be cocompact groups of isometries of hyperbolic space Hn of real dimension n, n ≥ 3. Let H_i ⊂ G_i be infinite index 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 one duality group. This generalizes (1). In particular, if n = 3, H_i could be any freely indecomposable subgroup of G_i. 4) H_i 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 H_i 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.