Pattern rigidity in hyperbolic spaces: duality and PD subgroups

Abstract

body { counter-reset: list; } ol li { list-style: none; } ol li:before { content: "("counter(list, decimal) ") "; counter-increment: list; } 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: 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 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.