Existence and classification of maximal growth distributions

This article tackles the problem of existence and classification of maximal growth distributions on smooth manifolds. We show that maximal growth distributions of rank$>2$ abide by a full $h$-principle in all dimensions. We make use of M. Gromov's higher order convex integration and, on the way, we establish a new criterion for checking ampleness of a differential relation. As a consequence we answer in the positive, for $k>2$, the long-standing open question posed by M. Kazarian and B. Shapiro more than 25 years ago in [14] of whether any parallelizable manifold admits a $k$-rank distribution of maximal growth. We also answer several related open questions. For completeness we show that the differential relation of maximal growth for rank-$2$ distributions is not ample in any ambient dimension. Non-ampleness of the Engel and the $(2,3,5)$-conditions follow as particular cases.