Finitely generated connected locally compact groups

Hofmann and Morris [6] proved that a locally compact connected group G has a finite subset generating a dense subgroup if and only if the weight w(G) of G does not exceed c , the cardinality of the continuum. The minimum cardinality of such a topological generating set is an invariant of the group, is denoted by σ(G), and is called the topological rank of G . For compact abelian groups of weight ≤ c , this number is 1. It was also shown there that for any compact connected group G of weight ≤ c , the invariant σ(G) ≤ 2. Cleary and Morris [3] observed that σ(R) = n+1, for n ≥ 1 and proved the surprising result that for any compact connected group G with w(G) ≤ c , the invariant σ(G×Rn) = n+ 1 for n ≥ 1. Here we extend their result significantly. For example, if G is a compact connected group of weight ≤ c and L is a nonsingleton connected Lie group, then σ(G × L) = σ(L). For more general locally compact groups we bound the topological rank of a group in terms of an associated Lie group.