Sharp Regularity for Certain Nilpotent Group Actions on the Interval

According to the classical Plante-Thurston Theorem, all nilpotent groups of C2-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 C1-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 Hölder regularity of the derivatives. Half of the proof relies on results on random paths on Heisenberg-like groups that are interesting by themselves. © 2013 Springer-Verlag Berlin Heidelberg.