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Non-commutative multivariable Reidemester torsion and the Thurston norm. (English) Zbl 1147.57014
S. Harvey introduced in [Topology 44, No. 5, 895–945 (2005; Zbl 1080.57019)] a function $$\delta_n$$ on the first cohomology group of a manifold, that gives the degree of the $$n$$-th order Alexander polynomial, defined from the rational derived series. On the other hand, in [Pac. J. Math. 230, No. 2, 271–296 (2007)], S. Friedl had related these higher order Alexander polynomials to Reidemeister torsions. This relationship is analyzed here and used to show that $$\delta_n$$ is a seminorm.
The key tool here is to consider the natural representation from $$\pi_1(M)$$ to $$K[t_1^{\pm 1},\cdots t_m^{\pm 1}]$$, namely the ring of Laurent polynomials on $$m$$ conmuting variables over an skew field $$K$$, where $$m$$ is the first Betti number of $$M$$, $$K$$ is defined from the group ring of $$\pi/\pi^n$$ localized by $$\pi^1/\pi^n$$, and $$\pi^n$$ denotes the $$n$$-th term of the rational derived series. This representation is acyclic and, by naturality, its Reidemeister torsion contains the information of the $$n$$-th Alexander polynomial corresponding to any element of $$H^1(M;\mathbb Z)$$. All these constructions need a careful use of noncomutative group and ring theory, and using them, the authors can prove that $$\delta_n$$ is a seminorm on $$H^1(M,\mathbb Z)$$.
As a corollary, the authors show that these invariants stabilize: for any 3-manifold there exists $$n_0$$ so that $$\delta_n=\delta_{n_0}$$ for all $$n > n_0$$. The paper ends with some examples that illustrate the use of this result to compute the Thurston norm on some links, because it is bounded below by Harvey’s invariants.

##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
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