Topological Properties of Almost Abelian Groups

TL;DR

This paper investigates the topological and algebraic properties of almost Abelian Lie groups, focusing on their discrete subgroups, compactness, and homotopy types, providing new insights into their structure.

Contribution

It establishes that all discrete subgroups of complex simply connected almost Abelian groups are finitely generated and characterizes the conditions for compactness of these groups and their subgroups.

Findings

Discrete subgroups are finitely generated

No complex connected almost Abelian group is compact

Conditions for compactness of subgroups

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.