On the cut locus of free, step two carnot groups

In this note, we study the cut locus of the free, step two Carnot groups Gk 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 works by Myasnichenko and Montanari and Morbidelli, by exhibiting sets of cut points Ck ⊂ Gk which, for k ≥ 4, are strictly larger than conjectured ones. While the latter were, respectively, smooth semi-algebraic sets of codimension Θ(k2) and semi-algebraic sets of codimension Θ(k), the sets Ck 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 Gk. Furthermore, we study the relation of the cut locus with the so-called abnormal set. In the low dimensional cases, it is known that Abn0 (Gk) = Cut0 (Gk)\Cut0 (Gk), 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 Ck. The question whether Ck coincides with the cut locus for k ≥ 4 remains open.