New properties of length-extremals in free step-2 rank-4 Carnot groups

Abstract In the free, step-2, rank-4 sub-Riemannian Carnot group $$\mathbb {F}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> , we give a clean expression for length-extremals, we provide an explicit equation for conjugate points, we relate it with the conjectured cut locus of the origin $$\operatorname {Cut}(\mathbb {F}_4)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>Cut</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Finally, we give some upper estimates for the cut-time of extremals.