Topological Properties of Almost Abelian Groups

Abstract

An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian subgroup. We show that all discrete subgroups of complex simply connected almost Abelian groups are finitely generated. The topology of connected almost Abelian Lie groups is studied by expressing each connected almost Abelian Lie group as a quotient of its universal covering group by a discrete normal subgroup. We then prove that no complex connected almost Abelian group is compact, and give conditions for the compactness of connected subgroups of such groups. Towards studying the homotopy type of complex connected almost Abelian groups, we investigate the maximal compact subgroups of such groups.