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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Invariant measures for affine foliations

Authors: William M. Goldman, Morris W. Hirsch and Gilbert Levitt
Journal: Proc. Amer. Math. Soc. 86 (1982), 511-518
MSC: Primary 57R30; Secondary 58F11, 58F18
MathSciNet review: 671227
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Abstract: A (transversely) affine foliation is a foliation with an atlas whose coordinate changes are locally affine. Such foliations arise naturally in the study of affine structures on manifolds. In this paper we prove that an affine foliation with nilpotent affine holonomy group always admits a nontrivial transverse measure. Two proofs are given: one for noncompact manifolds, and another, valid for compact manifolds with $ (G,X)$-foliations (not necessarily affine) having nilpotent holonomy. These results are applied to prove that a certain cohomology class on a compact affine manifold with nilpotent holonomy is nonzero. Examples are discussed.

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  • [1] E. Fédida and P. Furness,Feuilletages transversalement affines de codimenson 1, C. R. Acad. Sci. Paris 282 (1976), 825-827. MR 0418114 (54:6158)
  • [2] D. Fried, W. Goldman and M. Hirsch, Affine manifolds with nilpoltent holonomy, Comm. Math. Helv. 56 (1981) (to appear). MR 656210 (83h:53062)
  • [3] P. Furness and E. Fédida, Transversely affine foliations, Glasgow Math. J. 17 (1976), 106-111. MR 0420653 (54:8665)
  • [4] W. Goldman, Discontinuous groups and the Euler class, §1, Ph.D. Thesis, University of California, Berkeley, 1980.
  • [5] W. Goldman and M. Hirsch, Polynomial forms on affine manifolds, Pacific J. Math. (to appear). MR 671843 (84f:53026)
  • [6] -, Parallel characteristic class of affine manifolds, (in preparation).
  • [7] W. Gottschalk and G. Hedlund, Topological dyamics, Amer. Math. Soc. Colloq. Publ., vol. 36, Amer. Math. Soc., Providence, R.I., 1965.
  • [8] D. Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), 1-55. MR 0463429 (57:3380)
  • [9] R. Kulkarni, On the principle of uniformization, J. Differential Geom. 13 (1978), 109-138. MR 520605 (81k:53009)
  • [10] J. Plante, Foliations with measure-preserving holonomy, Ann. of Math. 102 (1975), 327-361. MR 0391125 (52:11947)
  • [11] -, Solvable group actions on the line, University of North Carolina, 1980 (preprint). MR 722145 (85c:14030)
  • [12] D. Ruelle and D. Sullivan, Currents, flows, and diffeomorphisms, Topology 14 (1975), 319-327. MR 0415679 (54:3759)
  • [13] B. Seke, Sur les structures transversalement affines des feuilletages de codimension un, Ann. Inst. Fourier (Grenoble) 30 (1980), 1-30. MR 576071 (82b:57023)
  • [14] D. Sullivan and W. Thurston, Manifolds with canonical coordinates: some examples, Inst. Hautes Etude Sci. Publ. Math., 1979 (preprint).
  • [15] W. Thurston, The geometry and topology of three-manifolds, Chapter 4, Princeton Univ. Press, Princeton, N.J., 1979 (preprint). MR 1435975 (97m:57016)
  • [16] R. A. Blumenthal, Transversely homogeneous foliations, Ann. Inst. Fourier (Grenoble) 29 (1979), 143-158. MR 558593 (81h:57011)

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