In this note, we study the cut locus of the free, step two Carnot groups G_k with k generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Myasnichenko - 2002] and [Montanari, Morbidelli - 2016], by exhibiting sets of cut points C_k ⊂G_k which, for k ≥ 4, are strictly larger than conjectured ones. While the latter were, respectively, smooth semi-algebraic sets of codimension Θ(k^2) and

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... lgebraic sets of codimension Θ(k), the sets C_k are semi-algebraic and have codimension 2, yielding the best possible lower bound valid for all k on the size of the cut locus of G_k. Furthermore, we study the relation of the cut locus with the so-called abnormal set. In the low dimensional cases, it is known that Abn_0(G_k) = Cut_0(G_k)∖Cut_0(G_k), k=2,3. For each k ≥ 4, instead, we show that the cut locus always intersects the abnormal set, and there are plenty of abnormal geodesics with finite cut time. Finally, and as a straightforward consequence of our results, we derive an explicit lower bound for the small time heat kernel asymptotics at the points of C_k. The question whether C_k coincides with the cut locus for k≥ 4 remains open.