Sharp regularity for certain nilpotent group actions on the interval

2011/08/26 by G. Castro, Gonzalo Castro, E. Jorquera +6 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #Amino Acid Enzymes and Metabolism #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #math.DS #math.GR #math.GT

paper · pdf · doi:10.48550/arxiv.1108.5223

Final version. To appear in Math. Annalen

arxiv created 2013/09/10 · arxiv updated 2013/09/11

Abstract

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.

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