JournalsggdVol. 9, No. 3pp. 831–889

Centralizers of C1C^1-contractions of the half line

  • Christian Bonatti

    Université de Bourgogne, Dijon, France
  • Églantine Farinelli

    Université de Bourgogne, Dijon, France
Centralizers of $C^1$-contractions of the half line cover

Abstract

A subgroup GDiff+1([0,1])G\subset \operatorname{Diff}^1_+([0,1]) is C1C^1-close to the identity if there is a sequence hnDiff+1([0,1])h_n\in \operatorname{Diff}^1_+([0,1]) such that the conjugates hnghn1h_n g h_n^{-1} tend to the identity for the C1C^1-topology, for every gGg\in G. This is equivalent to the fact that GG can be embedded in the C1C^1-centralizer of a C1C^1-contraction of [0,+)[0,+\infty) (see [6] and Theorem 1.1).

We first describe the topological dynamics of groups C1C^1-close to the identity. Then, we show that the class of groups C1C^1-close to the identity is invariant under some natural dynamical and algebraic extensions. As a consequence, we can describe a large class of groups GDiff+1([0,1])G\subset \operatorname{Diff}^1_+([0,1]) whose topological dynamics implies that they are C1C^1-close to the identity.

This allows us to show that the free group F2\mathbb F_2 admits faithful actions which are C1C^1-close to the identity. In particular, the C1C^1-centralizer of a C1C^1-contraction may contain free groups.

Cite this article

Christian Bonatti, Églantine Farinelli, Centralizers of C1C^1-contractions of the half line. Groups Geom. Dyn. 9 (2015), no. 3, pp. 831–889

DOI 10.4171/GGD/330