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Journal of Lie Theory 29 (2019), No. 2, 457--471
Copyright Heldermann Verlag 2019

The Topological Generating Rank of Solvable Lie Groups

Herbert Abels
Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

Gennady A. Noskov
Sobolev Institute of Mathematics, Pevtsova 13, 644099 Omsk, Russia


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 $A$.

Keywords: Lie group, solvable, nilpotent, metabelian, topological generators, generating rank.

MSC: 20E25

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