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

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-Włodarczyk factorization of birational toric maps in blow-ups and blow-downs (solution of the weak Oda conjecture).

arXiv comment13 pages