Sharp regularity for certain nilpotent group actions on the interval

According to the classical Plante-Thurston Theorem, all nilpotent groups of $C^2$-diffeomorphisms of the closed interval are Abelian. Using techniques coming from the works of Denjoy and Pixton, Farb and Franks constructed a faithful action by $C^1$-diffeomorphisms of $[0,1]$ for every finitely-generated, torsion-free, non-Abelian nilpotent group. In this work, we give a version of this construction that is sharp in what concerns the H\"older regularity of the derivatives. Half of the proof relies on results on random paths on Heisenberg-like groups that are interesting by themselves.