Codimension one locally free actions of solvable Lie groups

Abstract

Let $G$ be a non-unimodular solvable Lie group which is a semidirect product of $R^m$ and $R^n$. We consider a codimension one locally free volume preserving action of $G$ on a closed manifold. It is shown that, under some conditions on the group $G$, such an action is homogeneous. It is also shown that such a group $G$ has a homogeneous action if and only if the structure constants of $G$ satisfy certain algebraic conditions.