Expansions of the real field by discrete subgroups of 𝐺𝑙_{𝑛}(ℂ)

Hieronymi, Philipp; Walsberg, Erik; Xu, Samantha

doi:10.1090/proc/15382

Expansions of the real field by discrete subgroups of $\operatorname {Gl}_n(\mathbb {C})$
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by Philipp Hieronymi, Erik Walsberg and Samantha Xu

Proc. Amer. Math. Soc. 149 (2021), 2221-2233

DOI: https://doi.org/10.1090/proc/15382

Published electronically: March 1, 2021

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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 real number $\lambda$, or $(\mathbb {R},<,+,\cdot ,\Gamma )$ defines the set of integers. When $\Gamma$ is not virtually abelian, the second case holds.

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