# Arithmeticity of groups Z^n rtimes Z

### Indiana Univ. Math. J. 71 (2022) 4, 1797-1818.

We study when the group ℤ^n⋊ℤ is arithmetic, where ℤ acts on ℤ^n by a hyperbolic, semisimple A∈ GL(n,ℤ). We give 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 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.