Actions localement libres du groupe affine. (French) Zbl 0577.57010

Ghys, Etiènne

Invent. Math. 82, 479-526 (1985).

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

Keywords:

invariant volume form; group of affine transformations of \({\mathbb{R}}\); locally free actions; closed 3-dimensional manifold; structural stability; actions of fundamental groups of closed oriented surfaces; foliations

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 · doi:10.1016/0040-9383(80)90005-1
[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) · doi:10.1007/BFb0085358
[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) · doi:10.2307/1971026
[18]	[Pal] Palmeira, C.F.B.: Open manifolds foliated by planes. Ann. Math.107, 109-131 (1978) · Zbl 0382.57010 · doi:10.2307/1971256
[19]	[Pes] Pesin, Y.A.: characteristic Lyapounov exponents and smooth ergodic theory. Russ. Math. Surv.32, 54-114 (1977) · Zbl 0383.58011 · doi:10.1070/RM1977v032n04ABEH001639
[20]	[Pla 1] Plante, J.: Foliations with measure preserving holonomy. Ann. Math.102, 327-361 (1975) · Zbl 0314.57018 · doi:10.2307/1971034
[21]	[Pla 2] Plante, J.: Asymptotic properties of foliations. Comment. Math. Helv.47, 449-456 (1972) · Zbl 0254.57015 · doi:10.1007/BF02566817
[22]	[Pla 3] Plante, J.: Anosov flows. Am. J. Math.94, 729-754 (1972) · Zbl 0257.58007 · doi:10.2307/2373755
[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 · doi:10.1016/0040-9383(68)90021-9
[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 · doi:10.2307/1970633
[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 · doi:10.1007/BF02584795
[29]	[Sac] Sacksteder, R.: Foliations and pseudo-groups. Am. J. Math.87, 79-102 (1965) · Zbl 0136.20903 · doi:10.2307/2373226
[30]	[Ste] Sternberg, S.: LocalC n transformations of the real line. Duke Math. J.24, 97-102 (1957) · Zbl 0077.06201 · doi:10.1215/S0012-7094-57-02415-8
[31]	[Sul] Sullivan, D.: Discrete conformal groups and measurable dynamics. Bull. A.M.S.6, 57-73 (1982) · Zbl 0489.58027 · doi:10.1090/S0273-0979-1982-14966-7
[32]	[Thu] Thurston, W.: Geometry and topology of 3-manifolds. Princeton: Notes from Princeton University 1978