L2-invariants and 3–manifolds, II (Stefan Friedl)

Geometrisation

Let \( N \) be a compact, orientable 3–manifold. Say that \( N \) is prime if it cannot be decomposed as the connected sum of two 3–manifolds both not homeomorphic to \( \mathbb S^3 \). (By the Sphere Theorem this is equivalent to \( N \) not containing any sphere not bounding a 3-ball, and \( M \not= \mathbb S^2 \times \mathbb S^2 \).)

An embedded torus \( T \subset N \) is said to be essential if the induced map \( \pi_1(T) \to \pi_1(N) \) is injective. By Papakryakopoulos’ Loop Theorem, if \( N \) is prime this is equivalent to \( T \) not being the boundary of an embeded solid torus.

Geometrisation Theorem (Perelman, conjectured by W. Thurston): Let \( N \) be a prime 3–manifold. Then one of the following holds:

  1. \( N \) is Seifert fibered;
  2. \( N \) is hyperbolic;
  3. \( N \) contains an incompressible torus.

By induction, this implies that every prime 3–manifold contains a finite collection of incompressible tori which cuts it into « geometric pieces », i.e. manifolds with torus boundary which are either hyperbolic or Seifert fibered. For ou examples of knot exteriors we have:

  • As we saw, the exteriors of the trefoil of 51 are Seifert fibered;
  • The exterior of the figure-eight and of 52 are hyperbolic;
  • The exterior of a Whitehead double contains an essential torus, namely the boundary of the tubular neighbourhood of the original knot. For example this decomposes the exterior of the double of the trefoil into a Seifert fibered piece (the exterior of the trefoil) and an hyperbolic piece.

\( L^2 \)-invariants

\( L^2 \)-Betti numbers

Given a finite CW-complex \( X \) and a morphism \( \gamma: \pi_1(X) \to \Gamma \) for some discrete group \( \Gamma \) we obtain \( L^2 \)-Betti numbers \( b_n^{(2)}(X ; \gamma) \) from the complex
\[
C_*^{(2)}(X ; \gamma) = \ell^2(\Gamma) \otimes_{{\mathbb Z}[\pi_1(X)]} C_*(\widetilde X).
\]
If \( \Gamma \) is residually finite, i.e. there exists a sequence \( \Gamma = \Gamma_0 \supset \Gamma_1 \supset \cdots \) of finite-index, normal subgroups \( \Gamma_i \le \Gamma \) with \( \bigcup_i \Gamma_i = \{ 1 \} \) then an alternative definition is given by:

Lück Approximation Theorem: With the notation above, let \( X_i = \Gamma_i \backslash \widetilde X \). Then:
\[
b_n^{(2)}(X ; \gamma) = \lim_{i \to +\infty} \frac{b_n(X_i)} {|\Gamma/\Gamma_i|}.
\]

Using only this theorem it is possible to prove the following properties of the \( L^2 \)-Betti numbers:

  • If \( \Gamma = \langle x \rangle \cong {\mathbb Z} \) then \( b_n^{(2)}(X; \gamma) = b_n(X ; {\mathbb Q}(x)) \).
  • For any aspherical, prime compact 3–manifold \( N \) we have \( b_*(N ; \pi_1(N) = 0 \).

The proof of the second point follows if we can prove it for geometric manifolds. For Seifert manifolds the fundamental groups are extensions of Fuchsian groups by \( {\mathbb Z} \) and it is easy to find an explicit sequence of finite-index subgroup where \( b_1 \) grows sublinearly in the index.

For hyperbolic manifolds one can use the following deep theorem of Agol :

Theorem: If \( N \) is an hyperbolic manifold there exists a finite cover \( M \to N \) such that \( M \) is a surface bundle over \( \mathbb S^1 \).

The fundamental group is then an extension of \( {\mathbb Z} \) by a surface group, for which it is easy to find a sequence of finite-index subgroup whose first Betti numbers grows sublinearly.

\( L^2 \)-torsion

Suppose that \( N, \gamma \) is \( L^2 \)-acyclic, meaning that the Betti numbers \( b_n^{(2)}(N ; \Gamma) \) are all zero. Then it is possible to define the \( L^2 \)-torsion \( \tau^{(2)}(N ; \gamma) \), which is a positive real number (at least if \( \Gamma \) is residually finite).

In the case where the cellular complex is concentrated in one degree, say \( C_2^{(2)} \overset{\partial_2}{\to} C_1^{(2)} \), then the Reidemeister torsion is equal to the Fuglede–Kadison determinant: \( \tau^{(2)} = \det^{(2)}(\partial) \).

For 3–manifolds the universal cover is \( L^2 \)-acyclic, and Geometrisation allows to compute exactly the \( L^2 \)-torsion.

Theorem (Lück–Schick): Let \( N \) be a compact, prime 3–manifold and let \( N_1, \ldots, N_r \) be the hyperbolic pieces in the geometric decomposition of \( N \). Then:
\[
log \tau^{(2)}(N ; \pi_1(N)) = \frac 1 {6\pi} \sum_{i=1}^r \mathrm{vol}(N_i).
\]

The \( L^2 \)-Alexander invariant

Definition

Let \( N \) be a 3–manifold with morphisms \( \gamma: \pi_1(N) \to \Gamma \) and \( \phi: \pi_1(N) \to \langle x \rangle \cong {\mathbb Z} \). We say that the triple \( (N , \gamma, \phi) \) is admissible if \( \phi \) factors through \( \gamma \). In this setting, for \( t \in ]0, +\infty[ \) put:
\[
\tau^{(2)}(N ; \gamma, \phi)(t) = \tau^{(2)}(N ; g \mapsto t^{\phi(g)}\gamma(g)).
\]
The \( L^2 \)-Alexander invariant is the function \( t \mapsto \tau^{(2)}(N ; \gamma, \phi)(t) \). Note that for \( t \not\in{\mathbb Q} \) it is not obvious at all that this is positive, since for this we need to prove that a matrix with coefficients in the group ring \( {\mathbb R}[\Gamma] \) (as opposed to \( {\mathbb Z}[\Gamma] \) in the definition of \( L^2 \)-torsion) is of determinant class.

Baby case: \( \Gamma = {\mathbb Z} \)

Let \( \Gamma = \langle x \rangle \cong {\mathbb Z} \), and consider the admissible triple \( N, \gamma=\phi, \phi) \). For a matrix \( A \in M_{n \times n}({\mathbb R}[x, x^{-1}]) \) the Fuglede–Kadison determinant is given by the Mahler measure of the classical determinant. Namely, \( \det(A) \) is a polynomial in \( x, x^{-1} \) and we have:
\[
\begin{array}{cc}
\det^{(2)}(A) = m(\det(A)) & := \int_{|z|=1} \log|\det(A)(z)| \frac{dz} z \\
& = C\prod_{\det(A)(z_i) = 0} \max(1, |z_i|)
\end{array}
\]
where \( C \) is the leading coefficient of \( \det(A) \).

It is then easy to see that, in the simplest case where the complex is \( {\mathbb Z}[x, x^{-1}]^m \overset{A}{\to} {\mathbb Z}[x, x^{-1}]^n \), that:
\[
\tau^{(2)}(N; \phi, \phi) = Ct^n \prod_{\det(A)(z_i) = 0} \max(1, t^{-1}|z|_i).
\]
We see that \( \tau^{(2)}(N ; \phi, \phi) \) is a piecewise polynomial function, in particular \( \tau^{(2)}(N ; \phi, \phi) \equiv C \prod_i |z_i| \) in a neighbourhood of 0 and \( \equiv Ct^n \) in a neighbourhood of \( +\infty\).

In this setting the \( L^2 \)-Alexander invariant has many properties similar to the Reidemeister torsion.

  1. If \( N \) is a surface bundle over the circle and \( \phi \) is dual to the fiber then \( \tau^{(2)}(N ; \phi, \phi) \equiv 1 \) in a neighbourhood of \( 0 \) and \( \equiv t^k \) for some \( k \ge 0 \) at infinity.
  2. The « degree at infinity » \( k \) is smaller than the Thurson norm of \( \phi \).
  3. The \( L^2 \)-Alexander invariant is « symmetric », in the sense that \( \tau^{(2)}(N ; \phi, \phi)(t^{-1}) = t^k\tau^{(2)}(N ; \phi, \phi)(t) \).
  4. If \( N \) is Seifert fibered then \( \tau^{(2)}(N ; \phi, \phi)(t) = \max(1, t^{\| \phi \|_T}) \).

\( L^2 \)-Alexander invariant for the universal cover

From now on \( \gamma \) is the identity \( \pi_1(N) \to \pi_1(N) \) and we denote by \( \tau^{(2)}(N ; \phi ) \) the \( L^2 \)-Alexander function \( \tau^{(2)}(N ; \gamma, \phi) \).

Theorem (Dubois–Friedl–Lück): Suppose that \( N \) is a fiber bundle over the circle with fiber a surface \( F \) and monodromy \( f \), and \( \phi \) is the class dual to \( F \). Let \( \lambda \) be the dilatation of \( f \) (the largest modulus of an eigenvalue of \( f_*: H_1(F; {\mathbb C}) \to H_1(F; {\mathbb C}) \)). Then \( \tau^{(2)}(N ; \phi)(t) = 1 \) for \( t \le \lambda^{-1} \) and \( \tau^{(2)}(N ; \phi, \phi)(t) = t^{-\chi(F)} \) for \( t \ge \lambda \).

It is not known whether \( \tau^{(2)}(N; \phi) \) can be constant in an interval strictly larger than \( [0, \lambda^{-1}] \).

Other properties of this \( L^2 \)-Alexander function, similar to the ones we saw in other settings, are:

  1. \( \tau^{(2)}(N; \phi) \) is symmetric around \( t = 1 \).
  2. The Fox–Milnor theorem extends to the \( L^2 \)-Alexander function, but it is essentially useless in this setting since every nonnegative function can be written in the form \( t \mapsto f(t)f(t^{-1}) \).

Theorem (G. Hermann): If \( N \) is Seifert fibered then \( \tau^{(2)}(N; \phi)(t) = \max(1, t^{\|\phi\|_T}) \).

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