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 ${\tau }^{(2)}(N,\varphi )(t)$ associated to any real first cohomology class $\varphi$ of $N$ is represented by a function of a positive real variable $t$. The paper shows that ${\tau }^{(2)}(N,\varphi )$ is continuous, everywhere positive, and asymptotically monomial in both ends. Moreover, the degree of ${\tau }^{(2)}(N,\varphi )$ equals the Thurston norm of $\varphi$. 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 $\gamma :{\pi }_1(N)\to G$, the $L^2$--Alexander torsion ${\tau }^{(2)}(N,\gamma ,\varphi )$ 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 ${\tau }^{(2)}(N,\gamma ,\varphi )$. Moreover, the generalized degree is bounded by the Thurston norm of $\varphi$.

35 pages, references added

References in corpus (2)

• Twisting $L^2$-invariants with finite-dimensional representations
• Three flavors of twisted invariants of knots

Cited by in corpus (4)

• Twisting $L^2$-invariants with finite-dimensional representations
• Survey on approximating L^2-invariants by their classical counterparts: Betti numbers, torsion invariants and homological growth
• The $L^2$-Alexander torsion for Seifert fiber spaces
• Gluing formulas for the $L^2$-Alexander torsions