A family of matrix-tree multijections

For a natural class of $r \times n$ integer matrices, we construct non-convex polytope which periodically tiles $\mathbb R^n$. From this tiling, provide family geometrically meaningful maps from generalized sandpile group to set spanning trees give multijective proofs for several higher-dimensional matrix-tree theorems. In particular, these multijections can be applied graphs, regular matroids, cell complexes with torsion-free forest, and representable arithmetic matroids multiplicity one basis. This generalizes bijection given by Backman, Baker, Yuen extends work Duval, Klivans, Martin.