
Journal of Lie Theory 29 (2019), No. 2, 457471 Copyright Heldermann Verlag 2019 The Topological Generating Rank of Solvable Lie Groups Herbert Abels Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany abels@math.unibielefeld.de Gennady A. Noskov Sobolev Institute of Mathematics, Pevtsova 13, 644099 Omsk, Russia g.noskov@googlemail.com [Abstractpdf] 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) 123134]: 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 nontrivial isotypic components of the $\mathbb{R}Q$module $A$. Keywords: Lie group, solvable, nilpotent, metabelian, topological generators, generating rank. MSC: 20E25 [ Fulltextpdf (148 KB)] for subscribers only. 