UNIMODULAR GROUPS OF TYPE ℝ3 ⋊ ℝ Lee, Jong-Bum; Lee, Kyung-Bai; Shin, Joon-Kook; Yi, Seung-Hun;
There are 7 types of 4-dimensional solvable Lie groups of the form which are unimodular and of type (R). They will have left. invariant Riemannian metrics with maximal symmetries. Among them, three nilpotent groups are well known to have lattices. All the compact forms modeled on the remaining four solvable groups are characterized: (1) has lattices. For each lattice, there are infra-solvmanifolds with holonomy groups 1, . (2) Only some of , called , have lattices with no non-trivial infra-solvmanifolds. (3) does not have a lattice nor a compact form. (4) does not have a lattice, but has infinitely many compact forms. Thus the first Bieberbach theorem fails on . This is the lowest dimensional such example. None of these compact forms has non-trivial infra-solvmanifolds.