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
abels@math.uni-bielefeld.de
Gennady A. Noskov
Sobolev Institute of Mathematics, Pevtsova 13, 644099 Omsk, Russia
g.noskov@googlemail.com
[Abstract-pdf]
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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