Pattern Rigidity in Hyperbolic Spaces: Duality and PD Subgroups

TL;DR

This paper extends pattern rigidity results in hyperbolic spaces to a broad class of quasiconvex subgroups, including duality and Poincare duality groups, leading to new quasi-isometric rigidity conclusions for related graphs of groups.

Contribution

It generalizes Schwartz's pattern rigidity theorem to include various quasiconvex subgroups with specific limit set properties and applies these results to derive quasi-isometric rigidity for certain graphs of groups.

Findings

Pattern rigidity holds for broad classes of quasiconvex subgroups.

Extension of rigidity results to subgroups with complex limit sets.

Quasi-isometric rigidity for graphs of groups with specified vertex and edge groups.

Abstract

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 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 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)-(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 a result of Mosher-Sageev-Whyte, 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.