Abstract
We classify smooth locally free actions of the real affine group on closed orientable three-dimensional manifolds up to smooth conjugacy. As a corollary, there exists a nonhomogeneous action when the manifold is the unit tangent bundle of a closed surface with a hyperbolic metric.
-
[As1]
M. Asaoka, "Regular projectively Anosov flows on three-dimensional manifolds," Ann. Inst. Fourier $($Grenoble$)$, vol. 60, iss. 5, pp. 1649-1684, 2010.
@article {As1, MRKEY = {2766227},
AUTHOR = {Asaoka, Masayuki},
TITLE = {Regular projectively {A}nosov flows on three-dimensional manifolds},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {60},
YEAR = {2010},
NUMBER = {5},
PAGES = {1649--1684},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {37D30 (37C10 37E99 57R30)},
MRNUMBER = {2766227},
MRREVIEWER = {Shigenori Matsumoto},
DOI={10.5802/aif.2569},
ZBLNUMBER = {1202.37030},
} -
[Be2]
M. Belliart, "Locally free actions of solvable Lie groups and cohomology," Ergodic Theory Dynam. Systems, vol. 26, iss. 1, pp. 1-17, 2006.
@article {Be2, MRKEY = {2201934},
AUTHOR = {Belliart, Michel},
TITLE = {Locally free actions of solvable {L}ie groups and cohomology},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {26},
YEAR = {2006},
NUMBER = {1},
PAGES = {1--17},
ISSN = {0143-3857},
MRCLASS = {37C85 (37C15 43A80)},
MRNUMBER = {2201934},
MRREVIEWER = {Karel Dekimpe},
DOI = {10.1017/S0143385705000295},
ZBLNUMBER = {1087.37020},
} -
[Ca]
E. E. Cawley, "The Teichmüller space of an Anosov diffeomorphism of $T^2$," Invent. Math., vol. 112, iss. 2, pp. 351-376, 1993.
@article {Ca, MRKEY = {1213107},
AUTHOR = {Cawley, Elise E.},
TITLE = {The {T}eichmüller space of an {A}nosov diffeomorphism of {$T\sp 2$}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {112},
YEAR = {1993},
NUMBER = {2},
PAGES = {351--376},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {58F15},
MRNUMBER = {1213107},
MRREVIEWER = {Boris Hasselblatt},
DOI = {10.1007/BF01232439},
ZBLNUMBER={0788.58041},
} -
[Gh0] &. Ghys, Sur les actions localement libres du group affine, 1979.
@misc{Gh0,
author = {Ghys, {É}tienne},
TITLE = {Sur les actions localement libres du group affine},
YEAR={1979},
NOTE={Thèse de 3ème cycle, {L}ille},
} -
[Gh6]
&. Ghys, "Flots d’Anosov sur les $3$-variétés fibrées en cercles," Ergodic Theory Dynam. Systems, vol. 4, iss. 1, pp. 67-80, 1984.
@article {Gh6, MRKEY = {0758894},
AUTHOR = {Ghys, {É}tienne},
TITLE = {Flots d'{A}nosov sur les {$3$}-variétés fibrées en cercles},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {4},
YEAR = {1984},
NUMBER = {1},
PAGES = {67--80},
ISSN = {0143-3857},
MRCLASS = {58F15 (58F17)},
MRNUMBER = {0758894},
MRREVIEWER = {Alberto Verjovsky},
DOI = {10.1017/S0143385700002273},
ZBLNUMBER={0527.58030},
} -
[Gh]
&. Ghys, "Actions localement libres du groupe affine," Invent. Math., vol. 82, iss. 3, pp. 479-526, 1985.
@article {Gh, MRKEY = {0811548},
AUTHOR = {Ghys, {É}tienne},
TITLE = {Actions localement libres du groupe affine},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {82},
YEAR = {1985},
NUMBER = {3},
PAGES = {479--526},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {58F11 (22E40 30F35 57S25 58F18)},
MRNUMBER = {0811548},
MRREVIEWER = {Alberto Verjovsky},
DOI = {10.1007/BF01388867},
ZBLNUMBER = {0577.57010},
} -
[Gh5]
&. Ghys, "Flots d’Anosov dont les feuilletages stables sont différentiables," Ann. Sci. École Norm. Sup., vol. 20, iss. 2, pp. 251-270, 1987.
@article {Gh5, MRKEY = {0911758},
AUTHOR = {Ghys, {É}tienne},
TITLE = {Flots d'{A}nosov dont les feuilletages stables sont différentiables},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {20},
YEAR = {1987},
NUMBER = {2},
PAGES = {251--270},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {58F15},
MRNUMBER = {0911758},
MRREVIEWER = {Marlies Gerber},
URL = {http://www.numdam.org/item?id=ASENS_1987_4_20_2_251_0},
ZBLNUMBER = {0663.58025},
} -
[Gh1988]
&. Ghys, "Codimension one Anosov flows and suspensions," in Dynamical Systems, Valparaiso 1986, New York: Springer-Verlag, 1988, vol. 1331, pp. 59-72.
@incollection {Gh1988, MRKEY = {0961093},
AUTHOR = {Ghys, {É}tienne},
TITLE = {Codimension one {A}nosov flows and suspensions},
BOOKTITLE = {Dynamical Systems, {V}alparaiso 1986},
SERIES = {Lecture Notes in Math.},
VOLUME = {1331},
PAGES = {59--72},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1988},
MRCLASS = {58F15 (58F18 58F22)},
MRNUMBER = {0961093},
MRREVIEWER = {Shigenori Matsumoto},
DOI = {10.1007/BFb0083066},
ZBLNUMBER = {0672.58033},
} -
[Gh4]
&. Ghys, "Rigidité différentiable des groupes fuchsiens," Inst. Hautes Études Sci. Publ. Math., iss. 78, pp. 163-185 (1994), 1993.
@article {Gh4, MRKEY = {1259430},
AUTHOR = {Ghys, {É}tienne},
TITLE = {Rigidité différentiable des groupes fuchsiens},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {78},
YEAR = {1993},
PAGES = {163--185 (1994)},
ISSN = {0073-8301},
CODEN = {PMIHA6},
MRCLASS = {57M50 (20H10 30F99 57M60 57R30 58F15 58F18)},
MRNUMBER = {1259430},
MRREVIEWER = {Darryl McCullough},
URL = {http://www.numdam.org/item?id=PMIHES_1993__78__163_0},
ZBLNUMBER = {0812.58066},
} -
[Gh3]
&. Ghys, "Déformations de flots d’Anosov et de groupes fuchsiens," Ann. Inst. Fourier $($Grenoble$)$, vol. 42, iss. 1-2, pp. 209-247, 1992.
@article {Gh3, MRKEY = {1162561},
AUTHOR = {Ghys, {É}tienne},
TITLE = {Déformations de flots d'{A}nosov et de groupes fuchsiens},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {42},
YEAR = {1992},
NUMBER = {1-2},
PAGES = {209--247},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {58F18 (58F15 58F17)},
MRNUMBER = {1162561},
MRREVIEWER = {Takashi Tsuboi},
DOI={10.5802/aif.1290},
ZBLNUMBER = {0759.58036},
} -
[Gh2] &. Ghys.
@misc{Gh2,
author = {Ghys, {É}tienne},
NOTE={personal communication},
SORTYEAR={2012},
} -
[GS]
E. Ghys and V. Sergiescu, "Stabilité et conjugaison différentiable pour certains feuilletages," Topology, vol. 19, iss. 2, pp. 179-197, 1980.
@article {GS, MRKEY = {0572582},
AUTHOR = {Ghys, E. and Sergiescu, V.},
TITLE = {Stabilité et conjugaison différentiable pour certains feuilletages},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {19},
YEAR = {1980},
NUMBER = {2},
PAGES = {179--197},
ISSN = {0040-9383},
CODEN = {TPLGAF},
MRCLASS = {57R30},
MRNUMBER = {0572582},
MRREVIEWER = {Elmar Vogt},
DOI = {10.1016/0040-9383(80)90005-1},
ZBLNUMBER = {0478.57017},
} -
[KH] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge: Cambridge Univ. Press, 1995, vol. 54.
@book {KH, MRKEY = {1326374},
AUTHOR = {Katok, Anatole and Hasselblatt, Boris},
TITLE = {Introduction to the Modern Theory of Dynamical Systems},
SERIES = {Encycl. Math. Appl.},
VOLUME = {54},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1995},
PAGES = {xviii+802},
ISBN = {0-521-34187-6},
MRCLASS = {58Fxx (34Cxx 34Dxx 58-01 58F11 58F15)},
MRNUMBER = {1326374},
MRREVIEWER = {Edoh Amiran},
ZBLNUMBER = {0878.58020},
} -
[Li]
A. N. Livvsic, "Certain properties of the homology of $Y$-systems," Mat. Zametki, vol. 10, pp. 555-564, 1971.
@article {Li, MRKEY = {0293669},
AUTHOR = {Liv{š}ic, A. N.},
TITLE = {Certain properties of the homology of {$Y$}-systems},
JOURNAL = {Mat. Zametki},
FJOURNAL = {Akademiya Nauk SSSR. Matematicheskie Zametki},
VOLUME = {10},
YEAR = {1971},
PAGES = {555--564},
ISSN = {0025-567X},
MRCLASS = {58F10},
MRNUMBER = {0293669},
MRREVIEWER = {D. Newton},
ZBLNUMBER={0227.58006},
URL={http://mi.mathnet.ru/eng/mz/v10/i5/p555},
} -
[LMM]
R. de la Llave, J. M. Marco, and R. Moriyón, "Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation," Ann. of Math., vol. 123, iss. 3, pp. 537-611, 1986.
@article {LMM, MRKEY = {0840722},
AUTHOR = {de la Llave, R. and Marco, J. M. and Moriy{ó}n, R.},
TITLE = {Canonical perturbation theory of {A}nosov systems and regularity results for the {L}ivšic cohomology equation},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {123},
YEAR = {1986},
NUMBER = {3},
PAGES = {537--611},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {58F15 (34C35 58F30)},
MRNUMBER = {0840722},
MRREVIEWER = {Tong Ren Ding},
DOI = {10.2307/1971334},
} -
[LM2]
R. de la Llave and R. Moriyón, "Invariants for smooth conjugacy of hyperbolic dynamical systems. IV," Comm. Math. Phys., vol. 116, iss. 2, pp. 185-192, 1988.
@article {LM2, MRKEY = {0939045},
AUTHOR = {de la Llave, R. and Moriy{ó}n, R.},
TITLE = {Invariants for smooth conjugacy of hyperbolic dynamical systems. {IV}},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {116},
YEAR = {1988},
NUMBER = {2},
PAGES = {185--192},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {58F15},
MRNUMBER = {0939045},
MRREVIEWER = {Georgi{\u\i} Osipenko},
DOI={10.1007/BF01225254},
ZBLNUMBER = {0673.58038},
} -
[Ma] G. A. Margulis, On Some Aspects of the Theory of Anosov Systems, New York: Springer-Verlag, 2004.
@book {Ma, MRKEY = {2035655},
AUTHOR = {Margulis, Grigoriy A.},
TITLE = {On Some Aspects of the Theory of {A}nosov Systems},
SERIES = {Springer Monogr. Math.},
NOTE = {with a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2004},
PAGES = {vi+139},
ISBN = {3-540-40121-0},
MRCLASS = {37D20 (37C27 37C30 37C40)},
MRNUMBER = {2035655},
MRREVIEWER = {Boris Hasselblatt},
ZBLNUMBER = {1140.37010},
} -
[MM]
S. Matsumoto and Y. Mitsumatsu, "Leafwise cohomology and rigidity of certain Lie group actions," Ergodic Theory Dynam. Systems, vol. 23, iss. 6, pp. 1839-1866, 2003.
@article {MM, MRKEY = {2032491},
AUTHOR = {Matsumoto, S. and Mitsumatsu, Y.},
TITLE = {Leafwise cohomology and rigidity of certain {L}ie group actions},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {23},
YEAR = {2003},
NUMBER = {6},
PAGES = {1839--1866},
ISSN = {0143-3857},
MRCLASS = {37C85 (22F05 37D20)},
MRNUMBER = {2032491},
MRREVIEWER = {Meng-Kiat Chuah},
DOI = {10.1017/S0143385703000038},
ZBLNUMBER = {1052.22020},
} -
[Pl]
J. F. Plante, "Anosov flows, transversely affine foliations, and a conjecture of Verjovsky," J. London Math. Soc., vol. 23, iss. 2, pp. 359-362, 1981.
@article {Pl, MRKEY = {0609116},
AUTHOR = {Plante, J. F.},
TITLE = {Anosov flows, transversely affine foliations, and a conjecture of {V}erjovsky},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {The Journal of the London Mathematical Society. Second Series},
VOLUME = {23},
YEAR = {1981},
NUMBER = {2},
PAGES = {359--362},
ISSN = {0024-6107},
CODEN = {JLMSAK},
MRCLASS = {58F15},
MRNUMBER = {0609116},
MRREVIEWER = {Robert Roussarie},
DOI = {10.1112/jlms/s2-23.2.359},
ZBLNUMBER = {0465.58020},
} -
[Sa]
F. Salein, "Variétés anti-de Sitter de dimension 3 possédant un champ de Killing non trivial," C. R. Acad. Sci. Paris Sér. I Math., vol. 324, iss. 5, pp. 525-530, 1997.
@article {Sa, MRKEY = {1443988},
AUTHOR = {Salein, Fran{ç}ois},
TITLE = {Variétés anti-de {S}itter de dimension 3 possédant un champ de {K}illing non trivial},
JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
FJOURNAL = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
VOLUME = {324},
YEAR = {1997},
NUMBER = {5},
PAGES = {525--530},
ISSN = {0764-4442},
CODEN = {CASMEI},
MRCLASS = {53C50 (57S30)},
MRNUMBER = {1443988},
MRREVIEWER = {William Goldman},
DOI = {10.1016/S0764-4442(99)80384-X},
ZBLNUMBER = {0896.53043},
}