Separability of unipotent-free abelian subgroups in linear groups

TL;DR

The paper addresses the problem of separability for subgroups of linear groups, focusing on abelian subgroups that contain no unipotent elements in characteristic zero. It proves that any abelian unipotent-free subgroup of a finitely generated subgroup $\Gamma<\mathsf{GL}_n({\bf F})$ is separable, using a strategy that blends Grunewald-Segal specialization with Chevalley’s results to produce finite quotients, and a reduction to solvable subgroups within Borel subgroups. The argument relies on an inductive, case-based construction involving finite-ring homomorphisms and product maps into linear groups, culminating in a Lie-Kolchin-based reduction that transfers separability from a closed subgroup to the original group. These results apply to broad classes of linear groups, including subgroups of compact Lie groups, uniform lattices in semisimple Lie groups, Anosov and convex cocompact subgroups, and related quasi-isometric embeddings, while highlighting that the unipotent-free hypothesis is essential (as illustrated by the Baumslag-Solitar example).

Abstract

Let ${\bf F}$ be a field of characteristic zero. It is proved that for any finitely generated linear group $Γ<\mathsf{GL}_n({\bf F})$, every unipotent-free abelian subgroup of $Γ$ is separable.

Theorems & Definitions (12)

• Theorem 1
• Proposition 2
• proof
• Lemma 3
• Theorem 4
• proof
• Claim 5
• proof : Proof of Claim \ref{['claim']}
• proof : Proof of Theorem \ref{['main']}
• Remark 6