The Geometry of K-Orbits of a Subclass of MD5-Groups and Foliations Formed by their Generic K-Orbits

TL;DR

This paper explores the geometric structure and foliations of K-orbits in a specific subclass of MD5-groups, extending previous work by analyzing their geometry, maximal orbit foliations, and measurability.

Contribution

It provides a detailed description of the geometry of K-orbits and the associated foliations for a subclass of MD5-groups with a 3-dimensional derived ideal, building on prior classifications.

Findings

Description of K-orbit geometry for the subclass of MD5-groups.

Analysis of foliations formed by maximal dimension K-orbits.

Results on the measurability of these foliations.

Abstract

The present paper is a continuation of Le Anh Vu's ones [13], [14], [15]. Specifically, the paper is concerned with the subclass of connected and simply connected MD5-groups such that their MD5-algebras $\mathcal{G}$ have the derived ideal ${\mathcal{G}}^{1} : = [ \mathcal{G},\mathcal{G} ]\equiv$ ${\bf{R}}^{3}$. We shall describe the geometry of K-orbits of these MD5-groups. The foliations formed by K-orbits of maximal dimension of these MD5-groups and their measurability are also presented in the paper.