Semigroup Conjectures for Central Semidirect Product of Rⁿ with Rᵐ

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In this paper we prove two new results about closed semigroups in the family of solvable groups Hmn that are semidirect products of Rᵐ and Rⁿ, and for which the structure homomorphism maps nontrivially into the center of Aut(Rⁿ). The first result states that the closure of a semigroup generated by a set in Hmn that is not included in a maximal semigroup with nonempty interior is actually a group. The second result states that among the subsets of Hmn that are not included in a maximal proper semigroup, those that generate Hmn as a closed semigroup are dense. Results of this nature were obtained before only for extensions of nilpotent groups. Results of this nature were obtained before only for abelian and nilpotent Lie groups and their compact extensions. As an application of the technique developed in the paper, we find the minimal number of generators as a closed group and as a closed semigroup of Hmn.