Uniform rank metric stability of Lie algebras, Lie groups and lattices

Benjamin Bachner
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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 $F$.
李代数、李群和网格的均匀秩度量稳定性
我们研究离散群、李群和李代数在秩度量中的均匀稳定性,以及这些对象的均匀稳定性之间的联系。我们证明了半简单李代数远不是灵活地$\mathbb{C}$稳定的,半简单李群和高阶半简单李群中的格也不是严格地$\mathbb{C}$稳定的。此外,我们还证明了自由群在任何域$F$上都不是均匀柔性$F$稳定的。
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