Classification of 6-dimensional splittable flat solvmanifolds

A flat solvmanifold is a compact quotient $$\Gamma \backslash G$$ where G simply-connected solvable Lie group endowed with left invariant metric and $$ lattice of G. Any such can be written as $$G={\mathbb {R}}^k < imes _{\phi } {\mathbb {R}}^m$$ $${\mathbb the nilradical. In this article we focus on 6-dimensional splittable solvmanifolds, which are obtained quotienting by that decomposed =\Gamma _1 }\Gamma _2$$ , _1$$ lattices {R}}^k$$ respectively. We analyze relation between these conjugacy classes finite abelian subgroups $$\mathsf{GL}(n,{\mathbb {Z}})$$ known up to $$n\le 6$$ . From obtain classification solvmanifolds.