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E. Militon: Distortion elements in groups of smooth diffeomorphisms

E. Militon explained to the Groupe de travail de théorie ergodique in Orsay the following result.

Let $G$ be a group. Given $S\subset G$ and $g\in G$, the length $\ell_S(g)$ is the minimum length of a product of elements of $S\cup S^{-1}$ equal to $g$ (possibly $\infty$).

Definition. $g\in G$ is a distortion element if there exists a finite subset $S\subset G$, such that $\lim_{n\to\infty} \ell_S(g^n)/n = 0$.

Let $G=Diff^\infty_0(M)$ be the group of $C^\infty$-smooth diffeomorphisms of a compact manifold $M$ which, moreover, are isotopic to the identity. Let $d$ be a metric on $G$ which is compatible with the $C^\infty$-topology.

Theorem (Militon). If $g\in G$ is recurrent, i.e., $\liminf_{n\to\infty} d(g^n,Id)=0$, then $g$ is a distortion element.

Avila proved a similar result in the case $M=\mathbb S^1$ and Militon’s proof follows Avila’s.

Lemma 1. There exist two numerical sequences $\epsilon_n,k_n$ such that for any sequence of diffeomorphisms $h_n\in Diff^\infty_0(M)$ satisfying $d(h_n,Id)<\epsilon_n$ there exists a finite subset $S\subset G$ such that $\ell_S(h_n)\leq k_n$ for all $n\geq1$.

This Lemma is easily seen to imply the theorem. It is deduced from the next lemma using a non-trivial result on the decomposition of a diffeomorphism close enough to the identity into a composition of a bounded number of commutators of diffeomorphisms with small supports and themselves close to the identity.

$Diff^\infty_0(\mathbb R^d)$ denotes the set of smooth diffeomorphisms of $\mathbb R^d$ which are compact supported and are isotopic to the identity through a path of diffeomorphisms with supports all included in a fixed compact set of $\mathbb R^d$.

Lemma 2. There exist two numerical sequences $\epsilon_n,k_n$ such that for any pair of sequences of diffeomorphisms $f_n,g_n\in Diff^\infty_0(\mathbb R^d)$ satisfying $d(f_n,Id)<\epsilon_n$, $supp(f_n)\subset B(0,1)$ (and similar conditions on $g_n$), there exists a finite subset $S\subset G$ such that $\ell_S([f_n,g_n])\leq k_n$ for all $n\geq1$.