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Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation. (English) Zbl 1012.57022

The existence of a Reebless foliation in a compact 3-manifold \(M\) has strong implications on the topology of \(M\) (e.g. \(M\) is irreducible with infinite fundamental group and the universal covering of \(M\) is \(\mathbb R^3\); also, Thurston proposed an approach to the geometrization of such \(M\)). In the present paper, the first examples of closed hyperbolic 3-manifolds are given which contain no Reebless foliation. More precisely, it is shown that in each of the following three cases there are infinitely many closed hyperbolic 3-manifolds which contain: - no Reebless foliation; - no transversely oriented essential lamination; - neither a Reebless foliation nor a transversely oriented essential lamination but do contain essential laminations. It is expected (and a proof has recently been announced by Fenley) that some of the given examples are without any essential lamination. The examples [a subset of examples proposed by A. E. Hatcher, Ann. Inst. Fourier 42, No. 1-2, 313-325 (1992; Zbl 0759.57006)] are obtained by hyperbolic surgery (or Dehn filling) on once-punctured torus bundles over \(S^1\) (with pseudo-Anosov monodromy). The leaf space of any Reebless foliation of a 3-manifold \(M\), lifted to the universal covering of \(M\), is a simply-connected 1-manifold (second countable by not necessarily Hausdorff) on which the universal covering group induces a nontrivial action of \(\pi_1(M)\). Similarly, an essential lamination induces a nontrivial action of \(\pi_1(M)\) on a leaf space which is an \(\mathbb R\)-order tree. The main results are obtained by showing that the fundamental groups of the examples do not admit nontrivial actions on simply-connected 1-manifolds or \(\mathbb R\)-order trees, respectively. The examples, their relevant properties and the main ideas of the constructions are nicely described in the introductory section 2 of the paper. The main part of the paper is then a thorough analysis of group actions on non-Hausdorff 1-manifolds and \(\mathbb R\)-order trees.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57R30 Foliations in differential topology; geometric theory
57N10 Topology of general \(3\)-manifolds (MSC2010)

Citations:

Zbl 0759.57006
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References:

[1] Ian Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000), 431 – 449. · Zbl 0959.57009 · doi:10.2140/gt.2000.4.431
[2] Mark D. Baker, Covers of Dehn fillings on once-punctured torus bundles, Proc. Amer. Math. Soc. 105 (1989), no. 3, 747 – 754. · Zbl 0682.57006
[3] Mark D. Baker, Covers of Dehn fillings on once-punctured torus bundles. II, Proc. Amer. Math. Soc. 110 (1990), no. 4, 1099 – 1108. · Zbl 0731.57002
[4] Thierry Barbot, Actions de groupes sur les 1-variétés non séparées et feuilletages de codimension un, Ann. Fac. Sci. Toulouse Math. (6) 7 (1998), no. 4, 559 – 597 (French, with English and French summaries). · Zbl 0932.57027
[5] Sterling K. Berberian, Fundamentals of real analysis, Universitext, Springer-Verlag, New York, 1999. · Zbl 0914.26001
[6] S. Betley, J. H. Przytycki, and T. Żukowski, Hyperbolic structures on Dehn filling of some punctured-torus bundles over \?\textonesuperior , Kobe J. Math. 3 (1987), no. 2, 117 – 147. · Zbl 0633.57005
[7] Steven A. Bleiler and Craig D. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), no. 3, 809 – 833. · Zbl 0863.57009 · doi:10.1016/0040-9383(95)00040-2
[8] B. H. Bowditch, C. Maclachlan, and A. W. Reid, Arithmetic hyperbolic surface bundles, Math. Ann. 302 (1995), no. 1, 31 – 60. · Zbl 0830.57008 · doi:10.1007/BF01444486
[9] Mark Brittenham, Essential laminations in Seifert-fibered spaces, Topology 32 (1993), no. 1, 61 – 85. · Zbl 0791.57013 · doi:10.1016/0040-9383(93)90038-W
[10] Mark Brittenham, Ramin Naimi, and Rachel Roberts, Graph manifolds and taut foliations, J. Differential Geom. 45 (1997), no. 3, 446 – 470. · Zbl 0896.57017
[11] D. Calegari, Promoting essential laminations, I, preprint. · Zbl 1106.57014
[12] D. Calegari and N. Dunfield, Laminations and groups of homeomorphisms of the circle, Inv. Math., to appear. · Zbl 1025.57018
[13] A. Candel and L. Conlon, Foliations II, preprint. · Zbl 1035.57001
[14] I. Chiswell, Introduction to \(\Lambda\)-trees, World Scientific, Singapore, 2001. · Zbl 1004.20014
[15] E. Claus, Essential laminations in closed Seifert-fibered spaces, Thesis, University of Texas at Austin, 1991.
[16] Daryl Cooper, Craig D. Hodgson, and Steven P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, vol. 5, Mathematical Society of Japan, Tokyo, 2000. With a postface by Sadayoshi Kojima. · Zbl 0955.57014
[17] M. Culler, W. Jaco, and H. Rubinstein, Incompressible surfaces in once-punctured torus bundles, Proc. London Math. Soc. (3) 45 (1982), no. 3, 385 – 419. · Zbl 0515.57002 · doi:10.1112/plms/s3-45.3.385
[18] Marc Culler and John W. Morgan, Group actions on \?-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571 – 604. · Zbl 0658.20021 · doi:10.1112/plms/s3-55.3.571
[19] Marc Culler and Karen Vogtmann, A group-theoretic criterion for property \?\?, Proc. Amer. Math. Soc. 124 (1996), no. 3, 677 – 683. · Zbl 0865.20024
[20] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl. 11(1932), 333-375. · JFM 58.1124.04
[21] S. Fenley, Pseudo-Anosov flows and incompressible tori, preprint. · Zbl 1047.37017
[22] W. Floyd and A. Hatcher, Incompressible surfaces in punctured-torus bundles, Topology Appl. 13 (1982), no. 3, 263 – 282. · Zbl 0493.57004 · doi:10.1016/0166-8641(82)90035-9
[23] David Gabai, Taut foliations of 3-manifolds and suspensions of \?\textonesuperior , Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 193 – 208 (English, with French summary). · Zbl 0736.57010
[24] D. Gabai, Problems in foliations and laminations, Studies in Advanced Math. 2(2) (1997), 1-33. · Zbl 0888.57025
[25] David Gabai, Quasi-minimal semi-Euclidean laminations in 3-manifolds, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996) Int. Press, Boston, MA, 1998, pp. 195 – 242. · Zbl 0964.57015
[26] David Gabai and William H. Kazez, Order trees and laminations of the plane, Math. Res. Lett. 4 (1997), no. 4, 603 – 616. · Zbl 0887.57031 · doi:10.4310/MRL.1997.v4.n4.a14
[27] David Gabai and William H. Kazez, Group negative curvature for 3-manifolds with genuine laminations, Geom. Topol. 2 (1998), 65 – 77. · Zbl 0905.57011 · doi:10.2140/gt.1998.2.65
[28] David Gabai and Ulrich Oertel, Essential laminations in 3-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41 – 73. · Zbl 0685.57007 · doi:10.2307/1971476
[29] Sue E. Goodman, Closed leaves in foliations of codimension one, Comment. Math. Helv. 50 (1975), no. 3, 383 – 388. · Zbl 0318.57027 · doi:10.1007/BF02565757
[30] A. Haefliger and G. Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan, Ens. Math. 3 (1957), 107-125. · Zbl 0079.17101
[31] Allen Hatcher, Some examples of essential laminations in 3-manifolds, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 313 – 325 (English, with French summary). · Zbl 0759.57006
[32] Craig D. Hodgson, G. Robert Meyerhoff, and Jeffrey R. Weeks, Surgeries on the Whitehead link yield geometrically similar manifolds, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 195 – 206. · Zbl 0767.57007
[33] Troels Jørgensen, Compact 3-manifolds of constant negative curvature fibering over the circle, Ann. of Math. (2) 106 (1977), no. 1, 61 – 72. · Zbl 0368.53025 · doi:10.2307/1971158
[34] Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243 – 282. · Zbl 0947.57016 · doi:10.1007/s002220000047
[35] Joseph D. Masters, Virtual homology of surgered torus bundles, Pacific J. Math. 195 (2000), no. 1, 205 – 223. · Zbl 1019.57011 · doi:10.2140/pjm.2000.195.205
[36] Darryl McCullough, Automorphisms of punctured-surface bundles, Geometry and topology (Athens, Ga., 1985) Lecture Notes in Pure and Appl. Math., vol. 105, Dekker, New York, 1987, pp. 179 – 209. · Zbl 0607.57009
[37] John W. Morgan, \Lambda -trees and their applications, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 1, 87 – 112. · Zbl 0767.05054
[38] John W. Morgan and Peter B. Shalen, Degenerations of hyperbolic structures. II. Measured laminations in 3-manifolds, Ann. of Math. (2) 127 (1988), no. 2, 403 – 456. , https://doi.org/10.2307/2007061 John W. Morgan and Peter B. Shalen, Degenerations of hyperbolic structures. III. Actions of 3-manifold groups on trees and Thurston’s compactness theorem, Ann. of Math. (2) 127 (1988), no. 3, 457 – 519. · Zbl 0661.57004 · doi:10.2307/2007003
[39] J. Nielsen, Die Isomorphismengruppe der allgemeinen unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1917), 385-397. · JFM 46.0175.01
[40] S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965), 248 – 278 (Russian).
[41] Walter D. Neumann and Alan W. Reid, Arithmetic of hyperbolic manifolds, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 273 – 310. · Zbl 0777.57007
[42] Carlos Frederico Borges Palmeira, Open manifolds foliated by planes, Ann. Math. (2) 107 (1978), no. 1, 109 – 131. · Zbl 0382.57010
[43] Frédéric Paulin, Actions de groupes sur les arbres, Astérisque 241 (1997), Exp. No. 808, 3, 97 – 137 (French, with French summary). Séminaire Bourbaki, Vol. 1995/96.
[44] Józef H. Przytycki, Nonorientable, incompressible surfaces of genus 3 in \?_{\?}(\?\over\?) manifolds, Collect. Math. 34 (1983), no. 1, 37 – 79. · Zbl 0557.57005
[45] Józef H. Przytycki, Incompressibility of surfaces with four boundary components after Dehn surgery, Demonstratio Math. 17 (1984), no. 1, 119 – 126. · Zbl 0591.57007
[46] Alan W. Reid, A non-Haken hyperbolic 3-manifold covered by a surface bundle, Pacific J. Math. 167 (1995), no. 1, 163 – 182. · Zbl 0817.57014
[47] Alan W. Reid and Shicheng Wang, Non-Haken 3-manifolds are not large with respect to mappings of non-zero degree, Comm. Anal. Geom. 7 (1999), no. 1, 105 – 132. · Zbl 0930.57012 · doi:10.4310/CAG.1999.v7.n1.a4
[48] R. Roberts, J. Shareshian and M. Stein, in preparation.
[49] Rachel Roberts and Melanie Stein, Group actions on order trees, Topology Appl. 115 (2001), no. 2, 175 – 201. · Zbl 0985.57007 · doi:10.1016/S0166-8641(00)00060-2
[50] Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. · Zbl 0339.55004
[51] Paul A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math. (2) 100 (1974), 386 – 400. · Zbl 0295.57010 · doi:10.2307/1971077
[52] Peter B. Shalen, Dendrology of groups: an introduction, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 265 – 319. · Zbl 0649.20033 · doi:10.1007/978-1-4613-9586-7_4
[53] Peter B. Shalen, Dendrology and its applications, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 543 – 616. · Zbl 0843.20018
[54] Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. · Zbl 0548.20018
[55] W. Thurston, The geometry and topology of three-manifolds, Princeton, 1979.
[56] William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357 – 381. · Zbl 0496.57005
[57] William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417 – 431. · Zbl 0674.57008
[58] J. Weeks, Hyperbolic structures on three-manifolds, Thesis, Princeton University, 1985.
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