Degree of $$L^2$$ L 2 –Alexander torsion for 3–manifolds

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. Luck 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 \(\phi \).