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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 

Notes

Acknowledgments

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

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

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