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On groups generated by diffeomorphisms close to the identity. (Sur les groupes engendrés par des difféomorphismes proches de l’identité.) (French) Zbl 0809.58004
Let $$\Gamma$$ be a discrete group generated by $$S$$ and containing a free noncommutative subgroup and $$V$$ a compact real analytic manifold. We denote by $$\text{Diff}^ \omega (V)$$ its group of real analytic diffeomorphisms.
The author proves a version of the Zassenhaus lemma. There exists a neighbourhood $$U$$ of the identity in $$\text{Diff}^ \omega (V)$$ such that every morphism $$R : \Gamma \to \text{Diff}^ \omega (V)$$ which maps $$S$$ to $$U$$ is recurrent.
Interesting examples are given. The case of diffeomorphisms of the circle $$S^ 1$$ and the disk $$D^ 2$$ is studied in detail. We mention the following results. Each nilpotent subgroup of $$\text{Diff}^ \omega (S^ 1)$$ is commutative and each resoluble subgroup of $$\text{Diff}^ \omega (S^ 1)$$ has a commutative first derived group. A classification theorem of groups generated by real analytic diffeomorphisms of $$S^ 1$$ close to the identity is stated. For the closed disk $$D^ 2$$, the main result is: there exists a neighbourhood $$U$$ of the identity in $$\text{Diff}^ \omega (D^ 2)$$ such that if $$\Gamma$$ is generated by a subset of $$U$$ and the group induced by $$\Gamma$$ on $$S^ 1$$ is not resoluble, then the action of $$\Gamma$$ on $$D^ 2$$ is recurrent.

##### MSC:
 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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