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

TL;DR

This paper characterizes the definability of expansions of the real field by infinite discrete subgroups of GL$_n(\mathbb{C})$, showing they are either interdefinable with a multiplicative subgroup or define the integers, depending on the group's structure.

Contribution

It provides a classification of expansions of the real field by discrete subgroups of GL$_n(\mathbb{C})$, distinguishing between virtually abelian and non-virtually abelian cases.

Findings

If the subgroup is virtually abelian, the expansion is interdefinable with a multiplicative subgroup.

If not virtually abelian, the expansion defines the set of integers.

The result characterizes the logical complexity of these expansions.

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.