On the multiplication groups of three-dimensional topological loops

We clarify the structure of nilpotent Lie groups which are multiplication groups of $3$-dimensional simply connected topological loops and prove that non-solvable Lie groups acting minimally on $3$-dimensional manifolds cannot be the multiplication group of $3$-dimensional topological loops. Among the nilpotent Lie groups for any filiform groups ${\mathcal{F}}_{n+2}$ and ${\mathcal{F}}_{m+2}$ with $n,m>1$, the direct product ${\mathcal F}{n+2} \times \mathbb R$andthedirectproduct${\mathcal F}{n+2} \times Z {\mathcal F}{m+2}$withamalgamatedcenter$Z$ occur as the multiplication group of $3$-dimensional topological loops. To obtain this result we classify all $3$-dimensional simply connected topological loops having a $4$-dimensional nilpotent Lie group as the group topologically generated by the left translations.