Discrete Cocompact Subgroups of the Five-Dimensional Connected and Simply Connected Nilpotent Lie Groups

TL;DR

This paper classifies discrete cocompact subgroups of five-dimensional nilpotent Lie groups and describes their structure when the group has an Abelian factor, providing a comprehensive understanding of their subgroup composition.

Contribution

It determines all discrete cocompact subgroups of five-dimensional nilpotent Lie groups up to isomorphism and characterizes uniform subgroups in groups with Abelian factors.

Findings

Discrete cocompact subgroups classified up to isomorphism.

Uniform subgroups in groups with Abelian factors are direct products.

Provides explicit structure theorems for these subgroups.

Abstract

The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if $G=N\times A$ is a connected, simply connected, nilpotent Lie group with an Abelian factor $A$, then every uniform subgroup of $G$ is the direct product of a uniform subgroup of $N$ and ${\mathbb Z}^r$ where $r=\dim A$.