Measure theory in the geometry of $GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$

TL;DR

This paper constructs and proves the uniqueness of a measure invariant under the affine group over integers, providing a geometric framework for rational polyhedra using advanced algebraic geometry tools.

Contribution

It introduces a unique invariant measure on rational polyhedra under the affine integer group, connecting measure theory with the geometry of affine groups.

Findings

Established a G_n-invariant measure on rational polyhedra.

Proved the measure's uniqueness.

Applied the Morelli-W{ extlangle}odarczyk factorization in the proof.

Abstract

The $n$-dimensional affine group over the integers is the group $\mathcal G_n$ of all affinities on $\mathbb R^{n}$ which leave the lattice $ \mathbb Z^{n}$ invariant. $\mathcal G_n$ yields a geometry in the classical sense of the Erlangen Program. In this paper we construct a $\mathcal G_n$-invariant measure on rational polyhedra in $\mathbb R^n$, i.e., finite unions of simplexes with rational vertices in $\mathbb R^n$, and prove its uniqueness. Our main tool is given by the Morelli-W{\l}odarczyk factorization of birational toric maps in blow-ups and blow-downs (solution of the weak Oda conjecture).