Classification of 6-dimensional splittable flat solvmanifolds

TL;DR

This paper classifies 6-dimensional splittable flat solvmanifolds by analyzing conjugacy classes of finite order integer matrices, providing a detailed understanding of their structure and classification.

Contribution

It introduces a classification method for 6-dimensional splittable flat solvmanifolds using conjugacy classes of finite order matrices in dimensions 4 and 5.

Findings

Complete classification of 6-dimensional splittable flat solvmanifolds.

Identification of conjugacy classes of finite order matrices in dimensions 4 and 5.

Framework for understanding the structure of these solvmanifolds.

Abstract

A flat solvmanifold is a compact quotient $\Gamma\backslash G$ where $G$ is a simply-connected solvable Lie group endowed with a flat left invariant metric and $\Gamma$ is a lattice of $G$. Any such Lie group can be written as $G=\mathbb{R}^k\ltimes_{\phi} \mathbb{R}^m$ with $\mathbb{R}^m$ the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting $G$ by a lattice $\Gamma$ that can be decomposed as $\Gamma=\Gamma_1\ltimes_{\phi}\Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are lattices of $\mathbb{R}^k$ and $\mathbb{R}^m$, respectively. We obtain their classification by analyzing the conjugacy classes of integer matrices of finite order in dimensions 4 and 5.