Journal of the
Korean Mathematical Society JKMS

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group ${{\text{\rm Sol}_1}^{\kern-2pt 4}}$, and more generally, the crystallographic groups of ${{\text{\rm Sol}_1}^{\kern-2pt 4}}$. The maximal compact subgroup of $\operatorname{Isom}({{\text{\rm Sol}_1}^{\kern-2pt 4}})$ is $D_4=\mathbb Z_4\rtimes\mathbb Z_2$. We shall exhibit an infra-solvmanifold of ${{\text{\rm Sol}_1}^{\kern-2pt 4}}$ whose holonomy is $D_4$. This implies that all possible holonomy groups do occur; the trivial group, $\mathbb Z_2$ (5 families), $\mathbb Z_4$, $\mathbb Z_2\times\mathbb Z_2$ (5 families), and $\mathbb Z_4\rtimes\mathbb Z_2$ (2 families).

Keywords: solvmanifolds, infra-solvmanifolds, $\Sol$, $\Solof$, Bieberbach theorems, crystallographic groups

MSC numbers: Primary 20H15, 22E25, 20F16, 57S25