Journal of the Mathematical Society of Japan

Mapping tori with first Betti number at least two

Jack O. BUTTON

Full-text: Open access

Abstract

We show that given a finitely presented group G with β 1 ( G ) 2 which is a mapping torus Γ θ for Γ a finitely generated group and θ an automorphism of Γ then if the Alexander polynomial of G is non-constant, we can take β 1 ( Γ ) to be arbitrarily large. We give a range of applications and examples, such as any group G with β 1 ( G ) 2 that is F n -by- Z for F n the non-abelian free group of rank n is also F m -by- Z for infinitely many m . We also examine 3-manifold groups where we show that a finitely generated subgroup cannot be conjugate to a proper subgroup of itself.

Article information

Source
J. Math. Soc. Japan, Volume 59, Number 2 (2007), 351-370.

Dates
First available in Project Euclid: 1 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1191247591

Digital Object Identifier
doi:10.2969/jmsj/05920351

Mathematical Reviews number (MathSciNet)
MR2325689

Zentralblatt MATH identifier
1124.57001

Subjects
Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 57N10: Topology of general 3-manifolds [See also 57Mxx]

Keywords
mapping torus BNS invariant Alexander polynomial

Citation

BUTTON, Jack O. Mapping tori with first Betti number at least two. J. Math. Soc. Japan 59 (2007), no. 2, 351--370. doi:10.2969/jmsj/05920351. https://projecteuclid.org/euclid.jmsj/1191247591


Export citation

References

  • B. Baumslag and S. J. Pride, Groups with two more generators than relators, J. London Math. Soc., 17 (1978), 425–426.
  • G. Baumslag and P. B. Shalen, Amalgamated products and finitely presented groups, Comment. Math. Helv., 65 (1990) 243–254.
  • M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math., 129 (1997) 445–470.
  • R. Bieri, Homological dimension of discrete groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1976.
  • R. Bieri and R. Geoghegan, Connectivity properties of group actions on non-positively curved spaces, Mem. Amer. Math. Soc., 161 (2003), no.,765.
  • R. Bieri, W. D. Neumann and R. Strebel, A geometric invariant of discrete groups, Invent. Math., 90 (1987), 451–477.
  • R. Bieri and B. Renz, Valuations on free resolutions and higher geometric invariants of groups, Comment. Math. Helv., 63 (1988), 464–497.
  • A. Blass and P. M. Neumann, An application of universal algebra in group theory, Michigan Math. J., 21 (1974), 167–169.
  • M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschften, 319, Springer-Verlag, Berlin, 1999.
  • K. S. Brown, Trees, valuations, and the Bieri-Neumann-Strebel invariant, Invent. Math., 90 (1987), 479–504.
  • R. G. Burns, A. Karrass and D. Solitar, A note on groups with separable finitely generated subgroups, Bull. Austral. Math. Soc., 36 (1987), 153–160.
  • R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Ginn and Co., Boston, M. A., 1963.
  • N. M. Dunfield, Alexander and Thurston norms of fibered 3-manifolds, Pacific J. Math., 200 (2001), 43–58.
  • D. B. A. Epstein, Finite presentations of groups and 3-manifolds, Quart. J. Math. Oxford Ser., 12 (1961), 205–212.
  • M. Feighn and M. Handel, Mapping tori of free group automorphisms are coherent, Ann. of Math., 149 (1999), 1061–1077.
  • D. Fried and R. Lee, Realizing group automorphisms, Group actions on manifolds (Boulder, Colo. 1983), Contemp. Math. 36, Amer. Math. Soc., Providence, RI, 1985, pp.,427–432.
  • R. Geoghegan, M. L. Mihalik, M. Sapir and D. T. Wise, Ascending HNN extensions of finitely generated free groups are Hopfian, Bull. London Math. Soc., 33 (2001), 292–298.
  • J. Hempel, 3-manifolds, Ann. of Math. Studies No 86, Princeton University Press, Princeton, N. J., 1976.
  • W. Heil, On certain fiberings of $M_2\times S_1$, Proc. Amer. Math. Soc., 34 (1972), 280–286.
  • W. Heil, Some finitely presented non 3-manifold groups, Proc. Amer. Math. Soc., 53 (1975), 497–500.
  • W. Jaco, Surfaces embedded in $M^2\times S^1$, Canad. J. Math., 22 (1970), 553–568.
  • W. Jaco, Roots, relations and centralizers in 3-manifold groups, Geometric topology, Proc. Conf., Park City, Utah, 1974, Lecture Notes in Math., 438, Springer, Berlin, 1975, pp.,283–309.
  • I. Kapovich, A remark on mapping tori of free group endomorphisms, Preprint, available at http://front.math.ucdavis.edu/math.GR/0208189 (2002).
  • I. J. Leary, G. A. Niblo and D. T. Wise, Some free-by-cyclic groups, Groups St. Andrews 1997 in Bath, III, London Math. Soc. Lecture Note Ser. 261, Cambridge Univ. Press, Cambridge, 1999, pp.,512–516.
  • W. B. R. Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, 175, Springer-Verlag, New York, 1997.
  • C. T. McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup., 35 (2002), 153–171.
  • D. A. Neumann, 3-manifolds fibering over $S^1$, Proc. Amer. Math. Soc., 58 (1976), 353–356.
  • W. D. Neumann, Normal subgroups with infinite cyclic quotient, Math. Sci., 4 (1979), 143–148.
  • J. Stallings, On fibering certain 3-manifolds, Topology of 3-manifolds and related topics, Prentice-Hall, Englewood Cliffs, N. J., 1961, pp.,95–100.
  • W. P. Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc., 59 (1986), 99–130.
  • J. L. Tollefson, 3-manifolds fibering over $S^1$ with nonunique connected fiber, Proc. Amer. Math. Soc., 21 (1969), 79–80.
  • V. G. Turaev, The Alexander polynomial of a three-dimensional manifold, (Russian) Mat. Sb. (N.S.), 97 (139) (1975), 341–359, 463; (English) Math. USSR Sb., 26 (1975), 313–329.