Arithmeticity of groups ${\mathbb{Z}}^n⋊\mathbb{Z}$

Abstract

We study when the group ${\mathbb{Z}}^n{⋊}_A\mathbb{Z}$ is arithmetic where $A\in G{L}_n(\mathbb{Z})$ is hyperbolic and semisimple. We begin by giving a characterization of arithmeticity phrased in the language of algebraic tori, building on work of Grunewald-Platonov. We use this to prove several more concrete results that relate the arithmeticity of ${\mathbb{Z}}^n{⋊}_A\mathbb{Z}$ to the reducibility properties of the characteristic polynomial of $A$. Our tools include algebraic tori, representation theory of finite groups, Galois theory, and the inverse Galois problem.

22 pages. Comments welcome