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 $\Gamma$ of $G$ is strongly rigid in the sense of G. D. Mostow: If $\alpha:\Gamma\to G$ is a homomorphism of Lie groups such that $\alpha(\Gamma)$ is uniform in $G$, then there is an automorphism $\varphi$ of $G$ such that $\varphi\,|\,\Gamma=\alpha$. 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.