Classification of 6-dimensional splittable flat 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 $G={\mathbb{R}}^k{⋉}_{\varphi }{\mathbb{R}}^m$ 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 $\Gamma ={\Gamma }_1{⋉}_{\varphi }{\Gamma }_2$, 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.

arXiv commentCorrected examples 3.3 (A non-splittable lattice in a splittable Lie group) and 3.6 (A non almost abelian solvmanifold diffeomorphic to an almost abelian solvmanifold)