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=Rk ltimes_ϕ Rm with Rm 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=Γ1 ltimes_ϕ Γ2, where Γ1 and Γ2 are lattices of Rk and Rm, respectively. We analyze the relation between these lattices and the conjugacy classes of finite abelian subgroups of GL(n,Z), which is known up to n≤6. From this we obtain the classification of 6-dimensional splittable flat solvmanifolds.