Rigidity of lattices and syndetic hulls in solvable Lie groups

Abstract

First let $G$ be a completely solvable Lie group. We recall the proof of the following result: Any closed subgroup of $G$ possesses a unique syndetic hull in $G$. As a consequence we conclude that any uniform subgroup $Γ$ of $G$ is strongly rigid in the sense of G. D. Mostow: If $α:Γ\to G$ is a homomorphism of Lie groups such that $α(Γ)$ is uniform in $G$, then there is an automorphism $φ$ of $G$ such that $φ\,|\,Γ=α$. Now let $G$ be an arbitrary (exponential) solvable Lie group. We discuss certain conditions on closed subgroups of $G$ which are sufficient for the existence of a syndetic hull.