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 all filiform groups F n+2 and F m+2 with n, m > 1 , the direct product F n+2 R and the direct product F n+2 Z F m+2 with amalgamated center 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.