Rigidity of group actions on solvable Lie groups

Abstract.

We establish analogs of the three Bieberbach theorems for a lattice $\Gamma$ in a semidirect product $\mathsf{K}\rtimes\mathsf{K}$ where $\mathsf{S}$ is a connected, simply connected solvable Lie group and $\mathsf{K}$ is a compact subgroup of its automorphism group. We first prove that the action of $\Gamma$ on $\mathsf{S}$ is metrically equivalent to an action of $\Gamma$ on a supersolvable Lie group. The latter is shown to be determined by $\Gamma$ itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group $\Gamma$ as a lattice in a semidirect product $\mathsf{S}\rtimes\mathsf{F}$ with $\mathsf{F}$ being a finite group whose order is bounded by a constant only depending on the dimension of $\mathsf{S}$. This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup $\Upsilon \subset \mathsf{K}\rtimes\mathsf{K}$ that the quotient $\mathsf{S}/\Upsilon$ is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group.