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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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