Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation

Roberts, R.; Shareshian, J.; Stein, M.

doi:10.1090/S0894-0347-03-00426-0

Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation
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by R. Roberts, J. Shareshian and M. Stein;

J. Amer. Math. Soc. 16 (2003), 639-679

DOI: https://doi.org/10.1090/S0894-0347-03-00426-0

Published electronically: March 3, 2003

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Abstract:

We investigate group actions on simply-connected (second countable but not necessarily Hausdorff) 1-manifolds and describe an infinite family of closed hyperbolic 3-manifolds whose fundamental groups do not act nontrivially on such 1-manifolds. As a corollary we conclude that these 3-manifolds contain no Reebless foliation. In fact, these arguments extend to actions on oriented $\mathbb R$-order trees and hence these 3-manifolds contain no transversely oriented essential lamination; in particular, they are non-Haken.

References

Ian Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000), 431–449. MR 1799796, DOI 10.2140/gt.2000.4.431
Mark D. Baker, Covers of Dehn fillings on once-punctured torus bundles, Proc. Amer. Math. Soc. 105 (1989), no. 3, 747–754. MR 964452, DOI 10.1090/S0002-9939-1989-0964452-6
Mark D. Baker, Covers of Dehn fillings on once-punctured torus bundles. II, Proc. Amer. Math. Soc. 110 (1990), no. 4, 1099–1108. MR 1027087, DOI 10.1090/S0002-9939-1990-1027087-3
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). MR 1693597, DOI 10.5802/afst.911
Sterling K. Berberian, Fundamentals of real analysis, Universitext, Springer-Verlag, New York, 1999. MR 1654127, DOI 10.1007/978-1-4612-0549-4
S. Betley, J. H. Przytycki, and T. Żukowski, Hyperbolic structures on Dehn filling of some punctured-torus bundles over $S^1$, Kobe J. Math. 3 (1987), no. 2, 117–147. MR 908780
Steven A. Bleiler and Craig D. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), no. 3, 809–833. MR 1396779, DOI 10.1016/0040-9383(95)00040-2
B. H. Bowditch, C. Maclachlan, and A. W. Reid, Arithmetic hyperbolic surface bundles, Math. Ann. 302 (1995), no. 1, 31–60. MR 1329446, DOI 10.1007/BF01444486
Mark Brittenham, Essential laminations in Seifert-fibered spaces, Topology 32 (1993), no. 1, 61–85. MR 1204407, DOI 10.1016/0040-9383(93)90038-W
Mark Brittenham, Ramin Naimi, and Rachel Roberts, Graph manifolds and taut foliations, J. Differential Geom. 45 (1997), no. 3, 446–470. MR 1472884

Promoting essential laminations, I

Laminations and groups of homeomorphisms of the circle

Foliations II

Introduction to $\Lambda$-trees

Essential laminations in closed Seifert-fibered spaces

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. MR 1778789
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. MR 675414, DOI 10.1112/plms/s3-45.3.385
Marc Culler and John W. Morgan, Group actions on $\textbf {R}$-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571–604. MR 907233, DOI 10.1112/plms/s3-55.3.571
Marc Culler and Karen Vogtmann, A group-theoretic criterion for property $\textrm {FA}$, Proc. Amer. Math. Soc. 124 (1996), no. 3, 677–683. MR 1307506, DOI 10.1090/S0002-9939-96-03217-0

Sur les courbes définies par les équations différentielles à la surface du tore

11

Pseudo-Anosov flows and incompressible tori

W. Floyd and A. Hatcher, Incompressible surfaces in punctured-torus bundles, Topology Appl. 13 (1982), no. 3, 263–282. MR 651509, DOI 10.1016/0166-8641(82)90035-9
David Gabai, Taut foliations of $3$-manifolds and suspensions of $S^1$, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 193–208 (English, with French summary). MR 1162560, DOI 10.5802/aif.1289

Problems in foliations and laminations

2(2)

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. MR 1677889
David Gabai and William H. Kazez, Order trees and laminations of the plane, Math. Res. Lett. 4 (1997), no. 4, 603–616. MR 1470429, DOI 10.4310/MRL.1997.v4.n4.a14
David Gabai and William H. Kazez, Group negative curvature for $3$-manifolds with genuine laminations, Geom. Topol. 2 (1998), 65–77. MR 1619168, DOI 10.2140/gt.1998.2.65
David Gabai and Ulrich Oertel, Essential laminations in $3$-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41–73. MR 1005607, DOI 10.2307/1971476
Sue E. Goodman, Closed leaves in foliations of codimension one, Comment. Math. Helv. 50 (1975), no. 3, 383–388. MR 423371, DOI 10.1007/BF02565757
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
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). MR 1162564, DOI 10.5802/aif.1293
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. MR 1184411
Troels Jørgensen, Compact $3$-manifolds of constant negative curvature fibering over the circle, Ann. of Math. (2) 106 (1977), no. 1, 61–72. MR 450546, DOI 10.2307/1971158
Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243–282. MR 1756996, DOI 10.1007/s002220000047
Joseph D. Masters, Virtual homology of surgered torus bundles, Pacific J. Math. 195 (2000), no. 1, 205–223. MR 1781620, DOI 10.2140/pjm.2000.195.205
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. MR 873294
John W. Morgan, $\Lambda$-trees and their applications, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 1, 87–112. MR 1100579, DOI 10.1090/S0273-0979-1992-00237-9
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. MR 932305, DOI 10.2307/2007061

Die Isomorphismengruppe der allgemeinen unendlichen Gruppe mit zwei Erzeugenden

78

S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965), 248–278 (Russian). MR 200938
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. MR 1184416
Carlos Frederico Borges Palmeira, Open manifolds foliated by planes, Ann. of Math. (2) 107 (1978), no. 1, 109–131. MR 501018, DOI 10.2307/1971256
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. MR 1472536
Józef H. Przytycki, Nonorientable, incompressible surfaces of genus $3$ in $M_\varphi ({\lambda \over \mu })$ manifolds, Collect. Math. 34 (1983), no. 1, 37–79. MR 747857
Józef H. Przytycki, Incompressibility of surfaces with four boundary components after Dehn surgery, Demonstratio Math. 17 (1984), no. 1, 119–126. MR 760346
Alan W. Reid, A non-Haken hyperbolic $3$-manifold covered by a surface bundle, Pacific J. Math. 167 (1995), no. 1, 163–182. MR 1318168, DOI 10.2140/pjm.1995.167.163
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. MR 1674109, DOI 10.4310/CAG.1999.v7.n1.a4

in preparation

Rachel Roberts and Melanie Stein, Group actions on order trees, Topology Appl. 115 (2001), no. 2, 175–201. MR 1847462, DOI 10.1016/S0166-8641(00)00060-2
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, CA, 1976. MR 515288
Paul A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math. (2) 100 (1974), 386–400. MR 356086, DOI 10.2307/1971077
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. MR 919830, DOI 10.1007/978-1-4613-9586-7_{4}
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. MR 1170376
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7

The geometry and topology of three-manifolds

William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6

Hyperbolic structures on three-manifolds