L-torsion Invariants and the Magnus Representation of the Mapping Class Group

In this paper, we study a series of L-torsion invariants from the viewpoint of the mapping class group of a surface. We establish some vanishing theorems for them. Moreover we explicitly calculate the first two invariants and compare them with hyperbolic volumes. 1. Magnus representation Let Σg,1 be a compact oriented smooth surface of genus g with a boundary ∂Σg,1 ∼= S . In this paper, we always assume that g ≥ 1. We take and fix a base point ∗ ∈ ∂Σg,1 of Σg,1. Let Mg,1 be the mapping class group of Σg,1, namely, the group of all isotopy classes of orientation preserving diffeomorphisms of Σg,1 relative to the boundary. We denote π1(Σg,1, ∗) by Γ, which is a free group of rank 2g, and fix a generating system Γ = 〈x1, . . . , x2g〉. Let ZΓ be the group ring of Γ over Z. We write φ∗ ∈ Aut(Γ) to the automorphism induced from φ ∈ Mg,1. The following result, usually called the Dehn-Nielsen-Baer theorem, is classical and fundamental to study the mapping class group Mg,1 by using combinatorial group theories (see [9] Section 2.9). Proposition 1.1 (Zieschang [27]). The above induced homomorphism Mg,1 ∋ φ 7→ φ∗ ∈ Aut(Γ) is injective. As a corollary, we see that φ can be determined by the words φ∗(x1),. . . ,φ∗(x2g)∈ Γ. Since the fundamental formula γ = 1+ ∑2g i=1(∂γ/∂xi)(xi−1) holds in ZΓ for any γ ∈ Γ, the word φ∗(xj) is determined by {∂φ∗(xj)/∂xi}. Here ∂/∂xi : ZΓ → ZΓ denotes Fox’s free differential. See [1] Section 3.1 for a systematic treatment of the subject. The Magnus representation of the mapping class group is defined as follows. Definition 1.2. The Magnus representation of Mg,1 is defined by the assignment r : Mg,1 ∋ φ 7→ ( ∂φ∗(xj) ∂xi )