PUB - Publikationen an der Universität Bielefeld

We define the topological generating rank d (G) 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: 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/(G') over bar on A := (G') over bar is semisimple. Then d (G) is the maximum of the following two numbers: d (Q) and one plus the maximum of the multiplicities of the non-trivial isotypic components of the RQ-module A.