Expansions of the real field by discrete subgroups of Gl${}_n(\mathbb{C})$

Abstract

Let $\Gamma$ be an infinite discrete subgroup of Gl${}_n(\mathbb{C})$. Then either $(\mathbb{R},<,+,\cdot ,\Gamma )$ is interdefinable with $(\mathbb{R},<,+,\cdot ,{\lambda }^{\mathbb{Z}})$ for some $\lambda \in \mathbb{R}$, or $(\mathbb{R},<,+,\cdot ,\Gamma )$ defines the set of integers. When $\Gamma$ is not virtually abelian, the second case holds.