zbMATH — the first resource for mathematics

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.

58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
Full Text: DOI
[1] [A-G]d’Ambra, G., Gromov, M.: Lectures on transformation groups: geometry and dynamics. Preprint. · Zbl 0752.57017
[2] [Ar]Arnold, V.I.: Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Mir, Moscou, 1980.
[3] [Ba]Bavard, C.: Longueur stable des commutateurs. L’Ens. math. 37 (1991) 109-150. · Zbl 0810.20026
[4] [Bon]Bonatti, C.: Un point fixe commun pour des difféomorphismes commutants deS 2. Ann. Math. 129 (1989) 61-79. · Zbl 0689.57019 · doi:10.2307/1971485
[5] [Bo]Borel, A.: Linear algebraic groups. W.A. Benjamin, 1969. · Zbl 0186.33201
[6] [B-T]Broer, M., Tangerman, F.: From a differentiable to a real analytic perturbation theory, application to the Kupka-Smale theorems. Ergodic Th. & Dynam. Systems 6 (1986). · Zbl 0582.58019
[7] [E-T] Epstein, D.B.A., Thurston, W.P.: Transformation groups and natural bundles. Proc. London Math. Soc.38 (1979), 219-236. · Zbl 0409.58001 · doi:10.1112/plms/s3-38.2.219
[8] [Ha]Handel, M.: Commuting homeomorphisms ofS 2. Topology 31 (1992) 293-303. · Zbl 0755.57012 · doi:10.1016/0040-9383(92)90022-A
[9] [Ko]Koppel, N.: Commuting diffeomorphisms. Global Analysis, Berkeley 1968. Proc. Symp. Pure. math. 14 (1970) 165-184.
[10] [Le]Leslie, J.: On the group of real analytic diffeomorphisms of a compact real analytic manifold. Trans. A.M.S. 274 (1982). · Zbl 0526.58011
[11] [Ma]Margulis, G.A.: Discrete subgroups of semi-simple groups, Springer, Berlin Heidelberg New York, 1990.
[12] [M-R]Martinet, J., Ramis, J.P.: Classification analytique des équations différentelles non linéaires résonnantes du premier ordre. Ann. Scient. Ec. Norm. Sup. 16 (1983), 571-621.
[13] [Mt]Martinet, J.: Normalisation des champs de vecteurs holomorphes (d’après Brjuno). Séminaire Bourbaki, exposé 564, novembre 1980. LNM 901, Springer (1981) 55-70.
[14] [Na1]Nakai, I.: Separatrices for non solvable dynamics on C, O. Preprint.
[15] [Na2]Nakai, I.: A rigidity theorem for transverse dynamics. Preprint.
[16] [Nar]Narasimhan, R.: Analysis on real and complex manifolds. North Holland, 1968. · Zbl 0188.25803
[17] [Ne]Neumann, H.: Varieties of groups. Springer, Berlin, Heidelberg New York, 1967. · Zbl 0149.26704
[18] [Pl1]Plante, J.F.: Solvable groups acting on the line. Trans. A.M.S. 278 (1983) 401-414. · Zbl 0569.57012 · doi:10.1090/S0002-9947-1983-0697084-7
[19] [Pl2]Plante, J.F.: Subgroups of continuous groups acting differentiably on the half line. Ann. Inst. Fourier 34 (1984), 47-56. · Zbl 0519.57037
[20] [P-T] Plante, J.F., Thurston, W.P.: Polynomial growth in holonomy groups of foliations. Comment. Math. Helv. 51 (1976), 567-584. · Zbl 0348.57009 · doi:10.1007/BF02568174
[21] [Ra]Raghunathan, M.S.: Discrete subgroups of Lie groups. Springer, Berlin Heidelberg New York, 1972. · Zbl 0254.22005
[22] [Sa]Sacksteder, R.: Foliations and pseudogroups. Amer. J. Math. 87 (1965), 79-102. · Zbl 0136.20903 · doi:10.2307/2373226
[23] [We]Wehrfritz, B.A.F.: Infinite linear groups. Springer, Berlin Heidelberg New York, 1973.
[24] [Wi]Witte, D.: Arithmetic groups of higher ?-ranks cannot act on 1-manifolds. Preprint.
[25] [Zi]Zimmer, R.J.: Actions of semi-simple groups and discrete subgroups. Proc. Int. math. Cong. Berkeley, California, USA, 1986, pp. 1247-1258.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.