Sharp regularity for certain nilpotent group actions on the interval

Details Sharp regularity for certain nilpotent group actions on the interval Sharp regularity for certain nilpotent group actions on the interval

2011-08-26

arxiv.org/abs/1108.5223v2

Mathematics

Group Theory

34 pages, 5 figures. Comments are welcome

Scientific paper

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.

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