Inventiones mathematicae

, Volume 207, Issue 3, pp 981–1030 | Cite as

Degree of \(L^2\)–Alexander torsion for 3–manifolds

Article

Abstract

For an irreducible orientable compact 3-manifold N with empty or incompressible toral boundary, the full \(L^2\)–Alexander torsion \(\tau ^{(2)}(N,\phi )(t)\) associated to any real first cohomology class \(\phi \) of N is represented by a function of a positive real variable t. The paper shows that \(\tau ^{(2)}(N,\phi )\) is continuous, everywhere positive, and asymptotically monomial in both ends. Moreover, the degree of \(\tau ^{(2)}(N,\phi )\) equals the Thurston norm of \(\phi \). The result confirms a conjecture of J. Dubois, S. Friedl, and W. Lück and addresses a question of W. Li and W. Zhang. Associated to any admissible homomorphism \(\gamma :\pi _1(N)\rightarrow G\), the \(L^2\)–Alexander torsion \(\tau ^{(2)}(N,\gamma ,\phi )\) is shown to be continuous and everywhere positive provided that G is residually finite and \((N,\gamma )\) is weakly acyclic. In this case, a generalized degree can be assigned to \(\tau ^{(2)}(N,\gamma ,\phi )\). Moreover, the generalized degree is bounded by the Thurston norm of \(\phi \).

Mathematics Subject Classification

Primary 57M27 Secondary 57Q10 

References

  1. 1.
    Agol, I.: Criteria for virtual fibering. J. Topol. 1, 269–284 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Agol, I.: The virtual Haken conjecture, with an appendix by Agol, I., Groves, D., Manning, J. Doc. Math. 18, 1045–1087 (2013)Google Scholar
  3. 3.
    Aschenbrenner, M., Friedl, S., Wilton, H.: 3-Manifold Groups. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2015). ISBN:978-3-03719-154-5Google Scholar
  4. 4.
    Boyd, D.: Uniform approximation to Mahler’s measure in several variables. Can. Math. Bull. 41, 125–128 (1998)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Carey, A., Farber, M., Mathai, V.: Determinant lines, von Neumann algebras and \(L^2\) torsion. J. Reine Angew. Math. 484, 153–181 (1997)MathSciNetMATHGoogle Scholar
  6. 6.
    Cochran, T.: Noncommutative knot theory. Algebraic Geom. Topol. 4, 347–398 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dubois, J., Friedl, S., Lück, W.: The \(L^2\)–Alexander torsion of 3-manifolds. J. Topol. (2016). arXiv:1410.6918v3 (to appear)
  8. 8.
    Dubois, J., Friedl, S., Lück, W.: Three flavors of twisted invariants of knots. Introd. Modern Math. Adv. Lect. Math. 33, 143–170 (2015)MathSciNetGoogle Scholar
  9. 9.
    Dubois, J., Friedl, S., Lück, W.: The \(L^2\)–Alexander torsion is symmetric. Algebraic Geom. Topol. (2016). arXiv:1411.2292v1 (to appear)
  10. 10.
    Everest, G., Ward, T.: Heights of polynomials and entropy in algebraic dynamics. Springer, London (1999)CrossRefMATHGoogle Scholar
  11. 11.
    Friedl, S.: Twisted Reidemeister torsion, the Thurston norm and fibered manifolds. Geom. Dedicata 172, 135–145 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Friedl, S., Kim, T.: Twisted Alexander norms give lower bounds on the Thurston norm. Trans. Am. Math. Soc. 360, 4597–4618 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Friedl, S., Lück, W.: The \(L^2\)–torsion function and the Thurston norm of 3-manifolds, p. 22 (2015). arXiv:1510.00264v1 (preprint)
  14. 14.
    Friedl, S., Vidussi, S.: The Thurston norm and twisted Alexander polynomials, p. 17 (2012). arXiv:1204.6456v2 (preprint)
  15. 15.
    Harvey, S.: Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm. Topology 44, 895–945 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Harvey, S.: Monotonicity of degrees of generalized Alexander polynomials of groups and 3-manifolds. Math. Proc. Camb. Philos. Soc. 140, 431–450 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Herrmann, G.: The \(L^2\)–Alexander torsion for Seifert fiber spaces, pp. 11 (2016). arXiv:1602.08768v3 (preprint)
  18. 18.
    Li, W., Zhang, W.: An \(L^2\)-Alexander invariant for knots. Commun. Contemp. Math. 8, 167–187 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, W., Zhang, W.: An \(L^2\)–Alexander–Conway invariant for knots and the volume conjecture. Differential Geometry and Physics. Nankai Tracts Math. vol. 10, pp. 303–312. World Science Publisher, Hackensack (2006)Google Scholar
  20. 20.
    Liu, Y.: Virtual cubulation of nonpositively curved graph manifolds. J. Topol. 6, 793–822 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lott, J., Lück, W.: \(L^2\)-topological invariants of 3-manifolds. Invent. Math. 120, 15–60 (1995)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lück, W.: Approximating \(L^2\)-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4, 455–481 (1994)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lück, W.: \(L^2\)-Invariants: theory and applications to geometry and K-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (2002) (3. Folge)Google Scholar
  24. 24.
    Lück, W.: Twisting \(L^2\)–invariants with finite-dimensional representations, pp. 66 (2015). arXiv:1510.00057v1 (preprint)
  25. 25.
    McMullen, C.T.: The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology. Ann. Sci. Ec. Norm. Super. 35, 153–171 (2002). (4)MathSciNetMATHGoogle Scholar
  26. 26.
    Przytycki, P., Wise, D.T.: Graph manifolds with boundary are virtually special. J. Topol. 7, 419–435 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Przytycki, P., Wise, D.T.: Mixed 3-manifolds are virtually special, pp. 29 (2012). arXiv:1205.6742 (preprint)
  28. 28.
    Raimbault, J.: Exponential growth of torsion in abelian coverings. Algebraic Geom. Topol. 12, 1331–1372 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Thurston, W.P.: A norm for the homology of 3-manifolds. Mem. Am. Math. Soc. 59(339), 99–130 (1986)MathSciNetMATHGoogle Scholar
  30. 30.
    Turaev, V.: A homological estimate for the Thurston norm, pp. 32 (2002). arXiv:math.GT/0207267v1 (preprint)
  31. 31.
    Vidussi, S.: Norms on the cohomology of a 3-manifold and SW theory. Pac. J. Math. 208, 169–186 (2003)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Wise, D.T.: From Riches to RAAGs: 3-Manifolds, Right–Angled Artin Groups, and Cubical Geometry. CBMS Regional Conference Series in Mathematics (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchBeijingPeople’s Republic of China

Personalised recommendations