Semigroup Conjectures for Central Semidirect Product of R^n with R^m

Abstract

In this paper we prove two new results about closed semigroups in the family of solvable groups H_{mn} that are semidirect products of R^m and R^n, and for which the structure homomorphism maps nontrivially into the center of Aut(R^n). The first result states that the closure of a semigroup generated by a set in H_{mn} that is not included in a maximal semigroup with nonempty interior is actually a group. The second result states that among the subsets of H_{mn} that are not included in a maximal proper semigroup, those that generate H_{mn} 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 H_{mn}.