k-reduced groups and Milnor invariants

We characterize, in an algebraic and in a diagrammatic way, Milnor string link invariants indexed by sequences where any index appears at most $k$ times, for any fixed $k\ge 1$. The algebraic characterization is given in terms of an Artin-like action on the so-called $k$-reduced free groups; the diagrammatic characterization uses the langage of welded knot theory. The link case is also addressed.