×

zbMATH — the first resource for mathematics

Actions localement libres du groupe affine. (French) Zbl 0577.57010
In this interesting paper, locally free actions of the group GA of orientation preserving affine transformations of \({\mathbb{R}}\) on closed three-manifolds are considered. If M is a closed 3-dimensional manifold with \(H^ 1(M, {\mathbb{R}})=0\), then any locally free \(C^ 2\)-action of GA on M preserves a \(C^ 0\)-volume form (Theorem D). Any locally free \(C^ r\)-action (r\(\geq 2)\) of GA on any closed 3-manifold which preserves a \(C^ 0\)-volume form is \(C^{r-1}\)-conjugate to a ”homogeneous action”, i.e., an action of the form GA\(\times G/\Gamma \to G/\Gamma\), where G is a Lie group containing GA as a subgroup, \(\Gamma\) is a discrete uniform subgroup of G and the action is induced by left translations on G (Theorem B).
These are the main results of the article. However, it contains several other interesting results. Among the others: (1) homogeneous actions of GA on three manifolds are classified; (2) it is proved that any \(C^ 0\)- volume form preserved by a locally free \(C^ r\)-action of a non- unimodular Lie group G on a closed manifold M is of the class \(C^{r-2}\) if only dim M\(=\dim G+1\); (3) structural stability of some actions of fundamental groups of closed oriented surfaces of genus \(g\geq 2\) on \(S^ 1\) is established.
In the proofs, several deep results of the geometric theory of dynamical systems, ergodic theory and the theory of foliations are exploited.
Reviewer: P.Walczak

MSC:
57R30 Foliations in differential topology; geometric theory
57S20 Noncompact Lie groups of transformations
37A99 Ergodic theory
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [Ano] Anosov, D.V.: Geodesic flows on compact riemannian manifolds of negative curvature. Proc. Steklov Inst. Math., A.M.S. Translations 1969
[2] [Bob] Seke, B.: Thèse, Université de Strasbourg, 1982
[3] [Cha] Chatelet, G.: Sur les feuilletages induits par l’action de groupes de Lie nilpotents. Ann. Inst. Fourier27, 161-190 (1977) · Zbl 0349.57009
[4] [Dum] Duminy, G.: Bouts des feuilles dans les minimaux exceptionnels. (à paraître)
[5] [Eps] Epstein, D.B.A.: Ends, Topology of 3-manifolds and related topics. In: M. K. Fort, Proceed. of the University of Georgia Inst., Prentice Hall, 1961
[6] [Gel-Nai] Gelfand, I.M., Naimark, M.A.: Représentations unitaires du groupe des transformations linéaires de la droite. Dokl. Acad. Sci. URSS55, 7 (1947)
[7] [Ghy] Ghys, E.: Sur les actions localement libres du groupe affine. Thèse de 3ème cycle, Lille 1979
[8] [Ghy-Ser] Ghys, E., Sergiescu, V.: Stabilité et conjugaison différentiable pour certains feuilletages. Topology19, 179-197 (1980) · Zbl 0478.57017
[9] [Gol] Goldman, W.: Discontinuous groups and the Euler class. Doctoral Dissertion, Univ. of California, Berkeley, 1980
[10] [Gree] Green, L.W.: Remarks on uniformly expanding horocyclic foliations. J. Differ. Geom.13, 263 (1978)
[11] [Gre] Greenleaf, F.: Invariant means on topological groups. Math. Stud.16, American Book Company (1969) · Zbl 0174.19001
[12] [Hec 1] Hector, G.: On manifolds admitting locally free nilpotent Lie group actions of codimension 1 (à paraître)
[13] [Hec 2] Hector, G.: Feuilletages en cylindres. IIIème ELAM, Rio de Janeiro. Lect. Notes 597, 252-271 (1976)
[14] [Hec 3] Hector, G.: Croissance des feuilletages presque sans holonomie. Lect. Notes, School of Topology PUC, 1976 · Zbl 0393.57005
[15] [Hir-Pug-Shu] Hirsch, M., Pugh, C., Shub, M.: Invariant manifolds. Lect. Notes Math. 583 (1977)
[16] [Kir] Kirilov, A.: Eléments de la théorie des représentations. M.I.R. 1974
[17] [Mar] Marcus, B.: Ergodic properties of horocyclic flows for surfaces of negative curvature. Ann. Math.105, 81-105 (1977) · Zbl 0336.28008
[18] [Pal] Palmeira, C.F.B.: Open manifolds foliated by planes. Ann. Math.107, 109-131 (1978) · Zbl 0382.57010
[19] [Pes] Pesin, Y.A.: characteristic Lyapounov exponents and smooth ergodic theory. Russ. Math. Surv.32, 54-114 (1977) · Zbl 0383.58011
[20] [Pla 1] Plante, J.: Foliations with measure preserving holonomy. Ann. Math.102, 327-361 (1975) · Zbl 0314.57018
[21] [Pla 2] Plante, J.: Asymptotic properties of foliations. Comment. Math. Helv.47, 449-456 (1972) · Zbl 0254.57015
[22] [Pla 3] Plante, J.: Anosov flows. Am. J. Math.94, 729-754 (1972) · Zbl 0257.58007
[23] [Pla 4] Plante, J.: Locally free affine group actions. Trans. A.M.S.259, 449-456 (1980) · Zbl 0441.57022
[24] [Rat] Ratner, M.: Markov splitting forU-flows in three dimensional manifolds. Mat. Zametki6, 693-704 (1969)
[25] [Ros] Rosenberg, H.: Foliations by planes. Topology7, 131-138 (1968) · Zbl 0157.30504
[26] [Ros-Rou-Wei] Rosenberg, H., Roussarie, R., Weil, D.: A classification of 3-manifolds of rank two. Ann. Math.91, 449-469 (1970) · Zbl 0195.25404
[27] [Rue 1] Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. I.H.E.S.50, 27-50 (1979) · Zbl 0426.58014
[28] [Rue 2] Ruelle, D.: An inequality for the entropy of differentiable maps. Boll. Soc. Bras. Mat.9, 83-88 (1978) · Zbl 0432.58013
[29] [Sac] Sacksteder, R.: Foliations and pseudo-groups. Am. J. Math.87, 79-102 (1965) · Zbl 0136.20903
[30] [Ste] Sternberg, S.: LocalC n transformations of the real line. Duke Math. J.24, 97-102 (1957) · Zbl 0077.06201
[31] [Sul] Sullivan, D.: Discrete conformal groups and measurable dynamics. Bull. A.M.S.6, 57-73 (1982) · Zbl 0489.58027
[32] [Thu] Thurston, W.: Geometry and topology of 3-manifolds. Princeton: Notes from Princeton University 1978
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.