Defining R and G(R)
Group Theory
2021-03-31 v5 Logic
Abstract
We show that for Chevalley groups G(R) of rank at least 2 over a ring R the root subgroups are essentially (nearly always) the double centralizers of corresponding root elements. In very many cases this implies that R and G(R) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields. For such groups it then follows that the group G(R) is finitely axiomatizable in the appropriate class of groups provided R is finitely axiomatizable in the corresponding class of rings.
Cite
@article{arxiv.2004.13407,
title = {Defining R and G(R)},
author = {Dan Segal and Katrin Tent},
journal= {arXiv preprint arXiv:2004.13407},
year = {2021}
}
Comments
(1) New Theorem 1.1 generalizes earlier main theorems.(2) New version incorporates content of arXiv:2007.11440 (3) Latest version has small corrections. To appear in J. Eur. Math. Soc