The Topological Generating Rank of Solvable Lie Groups

H. Abels ; G. A. Noskov

Journal of Lie theory, Tome 29 (2019) no. 2, pp. 457-471

Résumé

We define the topological generating rank $d\left( G\right) $ of a connected Lie group $G$ as the minimal number of elements of $G$ needed to generate a dense subgroup of $G$. We answer the following question posed by K.\,H.\,Hofmann and S.\,A.\,Morris [see: {\it Finitely generated connected locally compact groups}, J. Lie Theory (formerly Sem. Sophus Lie) 2(2) (1992) 123--134]: What is the topological generating rank of a connected solvable Lie group? If $G$ is solvable we can reduce the question to the case that $G$ is metabelian. We can furthermore reduce to the case that the natural representation of $Q{:=}G^{ab}{:=} G/\overline{G^{\prime }}$ on $A:=\overline{G^{\prime}}$ is semisimple. Then $d\left(G\right)$ is the maximum of the following two numbers: $d\left(Q\right)$ and one plus the maximum of the multiplicities of the non-trivial isotypic components of the $\mathbb{R}Q$-module $A$.

Détail

Classification : 20E25
Mots-clés : Lie group, solvable, nilpotent, metabelian, topological generators, generating rank