Classification of 6-dimensional splittable flat solvmanifolds

Abstract

A flat solvmanifold is a compact quotient Γ\G where G is a simply-connected solvable Lie group endowed with a flat left invariant metric and Γ is a lattice of G. Any such Lie group can be written as G = R k ⋉ φ R m with R m the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting G by a lattice Γ that can be decomposed as Γ = Γ 1 ⋉ φ Γ 2 , where Γ 1 and Γ 2 are lattices of R k and R m , respectively. We obtain their classification by analyzing the conjugacy classes of integer matrices of finite order in dimensions 4 and 5.