Degree of $L^2$-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.~L\"uck and addresses a question of W.~Li and W.~Zhang. Associated to any admissible homomorphism $\gamma:\pi_1(N)\to G$, 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 $\phi$.