Let \( N \) be an irreducible, compact, orientable 3–manifold whose boundary is either empty or contains only tori as connected components. Call a triple \( (G, \gamma, \phi) \) admissible if \( G \) is a discrete group, \( \gamma : \pi_1(N) \to G \) and \( phi : \pi = \pi_1(N) \to {\mathbb Z} \) are homomorphisms such that there exists a commutative diagram:
\[
\begin{array}{ccc}
\pi_1(N) & \overset{\gamma}{\rightarrow} & G \\
& \underset{\phi}{\searrow} & \downarrow \\
& & \mathbb Z \end{array}
\]
Fix a cellulation of \( N \) and let \( C_*(\widetilde N) \) be the chain complex of the universal cover. Let \( t > 0 \) and define a representation:
\[
\kappa(\gamma, \phi, t) :\left\{ \begin{array}{ll}
{\mathbb Z} \pi \to {\mathbb R} G \\
g \mapsto t^{\phi(g)}\gamma(g)
\end{array} \right.
\]
with which the twisted \( L^2 \)-complex \( \ell^2(G) \otimes_{\kappa(\gamma, \phi, t)} C_*(\widetilde N) \). Let \( \tau^{(2)}(N; \gamma, \phi)(t) \) be the \( L^2 \)-torsion of this complex in the case where it is well-defined (when the complex is weakly acyclic and all its differentials of determinant class) and 0 otherwise. Consider this construction as associating to the admissible triple \( (G, \gamma, \phi) \) a function \( tau^{(2)}(N; \gamma, \phi): ]0, +\infty [ \to [0, +\infty[ \).

Remarks:

1. Changing the CW-structure on \( N \) changes the function \( tau^{(2)}(N; \gamma, \phi) \) by a factor \( t^r \) for some \( r \in {\mathbb R} \). It is possible to eleminate this indeterminacy by adding Euler structures to the mix.
2. If \( A \in M_{p\times p}({\mathbb Z}\pi) \) and \( C_* \) is the complex:
   \[
   (\ast) \qquad 0 \to ({\mathbb Z} \pi )^p \overset{A}{\rightarrow} ({\mathbb Z} \pi)^p \to 0
   \]
   then the \( L^2 \)-torsion of the twisted complex \( \ell^2(G) \otimes_{\kappa(\gamma, \phi, t)} C_* \) is equal to the « regularised determinant »
   \[
   \det{}^{\mathrm r}(A) = \left\{\begin{array}{ll}
   \det_{\mathcal NG}(\kappa(\gamma, \phi, t)(A)) \text{ if } \ker(\kappa(\gamma, \phi, t)(A)) = 0 ; \\
   0 \text{ otherwise.} \end{array} \right.
   \]

Examples

1. If \( G = \pi_1(N) \) and \( \gamma \) is the identity map, then for \( \phi \in H^1(N; {\mathbb R}) \) the \( L^2 \)-Alexander torsion is called the full \( L^2 \)-Alexander torsion and denoted \( \tau^{(2)}(N; \phi) \).
2. In the case where \( G = {\mathbb Z}^m \) is a free Abelian group then the Fuglede-Kadison determinant can be computed to be the Mahler measure of the classical determinant. In particular, when \( G = {\mathbb Z} \) and for the complex in \( (\ast) \) above the \( L^2 \)-Alexander torsion is given by:
   \[
   \tau^{(2)}(N; \phi, \phi) = Ct^d \prod_{i=1}^d \max(1, t^{-1}|z_i|)
   \]
   where the \( z_i \) are roots of the polynomial \( \det(\phi(A)) \) (see Stefan Friedl’s second lecture).

Multiplicative convexity

Return to the case where \( G \) is virtually cyclic. The, putting \( t = \log(u) \), in the case of an acyclic complex concentrated in one dimension we have that
\[
\log \tau^{(2)}(N; \phi, \phi)(e^u) = du + \sum_{i=1}^d \max(0, \log|z_i| – u) + \log (C)
\]
which is a piecewise affine function whose derivative is nondecreasing. In particular it is a convex function of \( u \), and this means that \( \tau^{(2)}(N; \phi, \phi) \) is a multiplicatively convex function. For a function \( f: ]0, +\infty[ \to [0, +\infty[ \) this means that for all \( t_0, t_1 > 0 \) we have
\[
\sqrt{f(t_0) f(t_1)} \ge f(\sqrt{t_0 t_1}).
\]
It implies that \( f \) is continuous, and that if it is zero at one point then it is identically zero. In the remainder we will explain how to prove multiplicative convexity for the full Alexander \( L^2 \)-torsion, using approximation by the Abelian case (using a theorem of Boyd on approximating the multivariate Mahler measure by univariate ones it is possible to pass from virtually cyclic to virtually f.g. abelian).

For this let \( 1 = \Gamma_0 \leftarrow \Gamma_1 \leftarrow \cdots \leftarrow \pi \) be a tower of finite quotients of \( \pi \) whose kernels have trivial intersection, and let \( G_n \) be a virtually free abelian quotient of \( \ker(\pi \to \Gamma_n) \), \( \gamma_n \) the morphism from this to \( G_n \) and \( \phi_n \) the lift of . Let \( A_n(t) = \kappa(\gamma_n, \phi, t)(A) \) and \( V_n(t) = \det^{\mathrm r} A_n(t) \). Let \( A_\infty(t) = \kappa(\mathrm{Id}, \phi, t)(A) \) and \( V_\infty(t) = \det^{\mathrm r} A_\infty(t) \). Fix \( t_0, t_1 > 0 \); we want to prove that
\[
\sqrt{V_\infty(t_0) V_\infty(t_1)} \ge V_\infty(\sqrt{t_0 t_1}).
\]
We know by the previous paragraph that
\[
\sqrt{V_n(t_0) V_n(t_1)} \ge V_n(\sqrt{t_0 t_1}).
\]
If we knew approximation for the residual chain we would be done, but this is not the case. We will use instead the two following properties of the regularised determinant:

1. \( \limsup_{n\to+\infty} V_n(t) \le V_\infty(t) \)
2. If \( \varepsilon > 0 \) and \( A \) is positive then \( \lim_{n\to+\infty} \det^{\mathrm r}(A_n(t) + \varepsilon) = \det^{\mathrm r}(A_\infty(t) + \varepsilon) \).
3. If \( A \) is positive then \( \lim_{\varepsilon \to 0} \det^{\mathrm r}(A_\infty(t) + \varepsilon) = \det^{\mathrm r}(A_\infty(t))

Using the first one we get that
\[
\sqrt{V_\infty(t_0) V_\infty(t_1)} \ge \limsup_{n\to+\infty}(V_n(\sqrt{t_0 t_1})).
\]
By adding \( \varepsilon \) to the matrices in the determinants on the right-hand and taking it to 0 in an intelligent way it is then possible to conclude.