Uniform rank metric stability of Lie algebras and groups

TL;DR

This paper investigates the stability properties of Lie algebras, Lie groups, and discrete groups under the rank metric, revealing limitations in their flexible and strict stability, especially for semisimple structures and free groups.

Contribution

It establishes new results on the non-stability of semisimple Lie algebras and groups, and free groups, in the context of the rank metric, highlighting their rigidity.

Findings

Semisimple Lie algebras are not flexibly $ ext{C}$-stable.

Semisimple Lie groups and their lattices are not strictly $ ext{C}$-stable.

Free groups are not uniformly flexibly $F$-stable over any field.

Abstract

We study uniform stability of discrete groups, Lie groups and Lie algebras in the rank metric, and the connections between uniform stability of these objects. We prove that semisimple Lie algebras are far from being flexibly $\mathbb{C}$-stable, and that semisimple Lie groups and lattices in semisimple Lie groups of higher rank are not strictly $\mathbb{C}$-stable. Furthermore, we prove that free groups are not uniformly flexibly $F$-stable over any field $F$.