Journal of the
Korean Mathematical Society JKMS

There are 7 types of 4-dimensional solvable Lie groups of the form $\mathbb R^3\rtimes_{\varphi} \mathbb R$ which are unimodular and of type (R). They will have left-invariant Riemannian metrics with maximal symmetries. Among them, three nilpotent groups ($\mathbb R^4$, ${\rm Nil}^3\times\mathbb R$ and ${\rm Nil}^4$) are well known to have lattices. All the compact forms modeled on the remaining four solvable groups ${\rm Sol}^3\times\mathbb R$, ${\rm Sol}^4_0$, ${\rm Sol'}_0^4$ and ${\rm Sol}_\lambda^4$ are characterized: (1) $\text{Sol}^3\times\mathbb R$ has lattices. For each lattice, there are infra-solvmanifolds with holonomy groups $1$, $\mathbb Z_2$ or $\mathbb Z_4$. (2) Only some of $\text{Sol}^4_{\lambda}$, called ${\rm Sol}_{m,n}^4$, have lattices with no non-trivial infra-solvmanifolds. (3) ${\rm Sol'}_0^4$ does not have a lattice nor a compact form. (4) ${\rm Sol}^4_0$ does not have a lattice, but has infinitely many compact forms. Thus the first Bieberbach theorem fails on ${\rm Sol}^4_0$. This is the lowest dimensional such example. None of these compact forms has non-trivial infra-solvmanifolds.

Keywords: Bieberbach Theorems, infra-homogeneous spaces, solvmanifold

MSC numbers: Primary 53C12; Secondary 53C20, 57R30