Inventiones mathematicae

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

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



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 



The author would like to thank Stefan Friedl and Wolfgang Lück for letting him learn their independent work and for subsequent valuable communications. The author also thanks Weiping Li for interesting conversations.


  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