P L representations of Anosov foliations

N. Hashiguchi

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 4, page 937-965
  • ISSN: 0373-0956

Abstract

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By choosing certain Birkhoff’s section to the geodesic flow of a negatively curved closed surface, E. Ghys showed that the unstable foliation of the geodesic flow has a transversely piecewise linear structure. We explicitly describe the holonomy homomorphism induced by this transversely piecewise linear structure and calculate its discrete Godbillon-Vey invariant.

How to cite

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Hashiguchi, N.. "$PL$ representations of Anosov foliations." Annales de l'institut Fourier 42.4 (1992): 937-965. <http://eudml.org/doc/74980>.

@article{Hashiguchi1992,
abstract = {By choosing certain Birkhoff’s section to the geodesic flow of a negatively curved closed surface, E. Ghys showed that the unstable foliation of the geodesic flow has a transversely piecewise linear structure. We explicitly describe the holonomy homomorphism induced by this transversely piecewise linear structure and calculate its discrete Godbillon-Vey invariant.},
author = {Hashiguchi, N.},
journal = {Annales de l'institut Fourier},
keywords = {Birkhoff's section to the geodesic flow of a negatively curved closed surface; unstable foliation; transversely piecewise linear structure; holonomy homomorphism; discrete Godbillon-Vey invariant},
language = {eng},
number = {4},
pages = {937-965},
publisher = {Association des Annales de l'Institut Fourier},
title = {$PL$ representations of Anosov foliations},
url = {http://eudml.org/doc/74980},
volume = {42},
year = {1992},
}

TY - JOUR
AU - Hashiguchi, N.
TI - $PL$ representations of Anosov foliations
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 4
SP - 937
EP - 965
AB - By choosing certain Birkhoff’s section to the geodesic flow of a negatively curved closed surface, E. Ghys showed that the unstable foliation of the geodesic flow has a transversely piecewise linear structure. We explicitly describe the holonomy homomorphism induced by this transversely piecewise linear structure and calculate its discrete Godbillon-Vey invariant.
LA - eng
KW - Birkhoff's section to the geodesic flow of a negatively curved closed surface; unstable foliation; transversely piecewise linear structure; holonomy homomorphism; discrete Godbillon-Vey invariant
UR - http://eudml.org/doc/74980
ER -

References

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  1. [CN] C. CAMACHO and A.L. NETO, Geometric Theory of Foliations, Birkhäuser, Boston, 1985. Zbl0568.57002
  2. [DS] G. DUMINY and V. SERGIESCU, Sur la nullité de l'invariant de Godbillon-Vey, C. R. Acad. Sci. Paris, 292 (1981), 821-824. Zbl0473.57015MR84a:57024
  3. [EM] S. EILENBERG and S. MACLANE, Relations between Homology and Homotopy Groups of Spaces, Ann. of Math., 46 (1945), 480-509. Zbl0061.40702MR7,137g
  4. [EHN] D. EISENBUD, U. HIRSCH and W. NEUMANN, Transverse foliations of Seifert bundles and self homeomorphism of the circle, Comment. Math. Helvetici, 56 (1981), 638-660. Zbl0516.57015MR83j:57016
  5. [F] D. FRIED, Transitive Anosov Flows and Pseudo-Anosov Maps, Topology, 22 (1983), 299-303. Zbl0516.58035MR84j:58095
  6. [Gh] E. GHYS, Sur l'invariance topologique de la classe de Godbillon-Vey, Ann. Inst. Fourier, 37, 4 (1987), 59-76. Zbl0633.58025MR89e:57023
  7. [GS] E. GHYS and V. SERGIESCU, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helvetici, 62 (1987), 185-239. Zbl0647.58009MR90c:57035
  8. [Gr] P. GREENBERG, Classifying spaces for foliations with isolated singularities, Trans. Amer. Math. Soc., 304 (1987), 417-429. Zbl0626.58030MR89a:57037
  9. [Ha1] N. HASHIGUCHI, On the Anosov diffeomorphisms corresponding to geodesic flows on negatively curved closed surfaces, J. Fac. Sci. Univ. Tokyo, 37 (1990), 485-494. Zbl0729.58040MR91k:58105
  10. [Ha2] N. HASHIGUCHI, On the rigidity of PL representations of a surface group, preprint. Zbl0829.57016
  11. [He] M. HERMAN, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Pub. Math., 49 (1978), 5-234. Zbl0448.58019MR81h:58039
  12. [Ma] B. MASKIT, On Poincaré's theorem for fundamental polygons, Adv. in Math., 7 (1971), 219-230. Zbl0223.30008MR45 #7049
  13. [Mi] J. MILNOR, On the 3-dimensional Brieskorn manifold M (p, q, r), in Knots, Groups and 3-Manifolds, Ann. of Math. Studies, 84 (1975), 175-225, Princeton. Zbl0305.57003MR54 #6169
  14. [T] T. TSUBOI, Area functionals and Godbillon-Vey cocycles, Ann. Inst. Fourier, 42, 1-2 (1992), 421-447. Zbl0759.57019MR94g:57032

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