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

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