Basic geometry of the affine group over Z

Abstract

The subject matter of this paper is the geometry of the affine group over the integers, $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$. Turing-computable complete $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-orbit invariants are constructed for angles, segments, triangles and ellipses. In rational affine $\mathsf{GL}(n,\mathbb Q)\ltimes \mathbb Q^n$-geometry, ellipses are classified by the Clifford--Hasse--Witt invariant, via the Hasse-Minkowski theorem. We classify ellipses in $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-geometry combining results by Apollonius of Perga and Pappus of Alexandria with the Hirzebruch-Jung continued fraction algorithm and the Morelli-W\l odarczyk solution of the weak Oda conjecture on the factorization of toric varieties. We then consider {\it rational polyhedra}, i.e., finite unions of simplexes in $\mathbb R^n$ with rational vertices. Markov's unrecognizability theorem for combinatorial manifolds states the undecidability of the problem whether two rational polyhedra $P$ and $P'$ are continuously $\mathsf{GL}(n,\mathbb Q)\ltimes \mathbb Q^n$-equidissectable. The same problem for the continuous $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-equi\-dis\-sect\-ability of $P$ and $P'$ is open. We prove the decidability of the problem whether two rational polyhedra $P,Q$ in $\mathbb R^n$ have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-orbit.