UNIMODULAR GROUPS OF TYPE ℝ3⋊ ℝ

Abstract

There are 7 types of 4-dimensional solvable Lie groups of the form <TEX>${\mathbb{R}^3}\;{\times}_{\varphi}\;{\mathbb{R}}$</TEX> which are unimodular and of type (R). They will have left. invariant Riemannian metrics with maximal symmetries. Among them, three nilpotent groups <TEX>$({\mathbb{R}^4},\;Nil^3\;{\times}\;{\mathbb{R}\;and\;Nil^4)$</TEX> are well known to have lattices. All the compact forms modeled on the remaining four solvable groups <TEX>$Sol^3\;{\times}\;{\mathbb{R}},\;Sol_0^4,\;Sol_0^'4\;and\;Sol_{\lambda}^4$</TEX> are characterized: (1) <TEX>$Sol^3\;{\times}\;{\mathbb{R}}$</TEX> has lattices. For each lattice, there are infra-solvmanifolds with holonomy groups 1, <TEX>${\mathbb{Z}}_2\;or\;{\mathbb{Z}}_4$</TEX>. (2) Only some of <TEX>$Sol_{\lambda}^4$</TEX>, called <TEX>$Sol_{m,n}^4$</TEX>, have lattices with no non-trivial infra-solvmanifolds. (3) <TEX>$Sol_0^{'4}$</TEX> does not have a lattice nor a compact form. (4) <TEX>$Sol_0^4$</TEX> does not have a lattice, but has infinitely many compact forms. Thus the first Bieberbach theorem fails on <TEX>$Sol_0^4$</TEX>. This is the lowest dimensional such example. None of these compact forms has non-trivial infra-solvmanifolds.