The Agrachev–Barilari–Boscain Method and Estimates for the Number of Segments of Horizontal Broken Lines Joining Points in the Canonical Carnot Group $G_{3,3}$

Using a generalization of the Agrachev–Barilari–Boscain method for proving the Rashevskii–Chow theorem, we estimate the minimum number $\mathcal {N}_{G_{3,3}}$ of segments of horizontal broken lines joining two arbitrary points on the six-dimensional two-step canonical Carnot group $G_{3,3}$ with corank $3$ horizontal distribution. We prove that $\mathcal {N}_{G_{3,3}}=3$.