TheL²-Alexander torsion of 3-manifolds

Abstract

We introduce L 2 -Alexander torsions for 3-manifolds, which can be viewed as a generalization of the L 2 -Alexander invariant of Li-Zhang. We state the L 2 -Alexander torsions for graph manifolds and we partially compute them for fibered manifolds. We furthermore show that, given any irreducible 3-manifold there exists a coefficient system such that the corresponding L 2 -torsion detects the Thurston norm.

Universal L2-torsion, polytopes and applications to 3-manifolds

“…In Section 3.4 we will see that the universal ‐torsion of a 3‐manifold determines the ‐torsion and more generally the ‐torsion function (also called ‐Alexander polynomial and ‐Alexander torsion) that recently was intensively studied, see for example, . The former invariant is determined by the volumes of the hyperbolic pieces in the Jaco–Shalen–Johannson decomposition of , see [, Theorem 0.7].…”

Twisting L2-invariants with finite-dimensional representations

“…Then the answer to Question 9.11 is positive. This is proven for Question 9.11 (1) in [11,Corollary 2.7]. Moreover, the answer for Question 9.11 (2) Remark 9.12 (A bound on the order of finite subgroups is necessary).…”

“…The following function is of interest and versions of it have already been studied in low-dimensions in [11,9,10,12,13,28,29,30]. Recall that we have made η φ * Ct more explicit in Example 1.5.…”

“…More generally, one can compute it for any graph manifold, see [11,Theorem 1.2]. If M i is hyperbolic, one knows at least…”

“…-torsion function appears and is investigated for compact connected orientable irreducible 3-manifolds M with infinite fundamental group and empty or toroidal boundary in [11,9,10], where a multiplicative version is used and called L 2 -Alexander torsion. In Friedl-Lück [16] and other forthcoming papers we will relate the degree of ρ (2) ( M ; φ) and more generally of ρ…”