Hyperbolic manifolds

Let \( \Sigma \) be a surface and \( f \in \mathrm{Homeo}^+(\Sigma) \). Let \( M \) be the 3–manifold obtained from \( \Sigma \times [0, 1] \) by identifying \( \Sigma \times \{0\} \) with \( \Sigma \times\{1\} \) via \( f \). If \( f \) is a pseudo-Anosov diffeomorphism then \( M \) is hyperbolic. If a 3–manifold \( M \) is obtained from this construction say that it is fibered. A theorem of Agol states that every hyperbolic manifold has a finite cover which is fibered.

If \( M \) is fibered with fiber \( \Sigma \) and monodromy \( f \) then its fundamental group has a splitting:
\[
1 \to \pi_1(\Sigma) \to \pi_1(M) \to {\mathbb Z} \to 1
\]
coming from the presentation
\[
\pi_1(M) = \langle \pi_1(\Sigma), t | \forall x \in \pi_1(\Sigma) t^{-1}xt = f_*(x) \rangle.
\]
More generally, if \( H \) is a group and \( f : H \to H \) is an injective morphism then the group obtained by:
\[
G = \langle H, t | \forall x \in H, t^{-1}xt = f(x) \rangle
\]
is called an ascending HNN-extension and denoted by \( H *_f \). Then:

• Any semi-direct product \( H \times {\mathbb Z} \) is an ascending HNN-extension;
• If \( G = H *_f \) let \( \phi: G \to {\mathbb Z} \) be the mosphism defined by \( \phi|_G \equiv 0 \) and \( \phi(t) = 1 \); it will be called the induces character of the extension \( H *_f \).

Definition: Let \( G \) be a group with a finite generating set \( S \). The Bieri–Neumann–Strebel invariant is the subset \( \Sigma(G) \subset H^1(G, {\mathbb R}) \setminus \{0\} \) containing all classes \( \phi \) such that the subgraph of the Cayley graph of \( G \) induced by the subset \( \{ g \in G: \phi(g) \ge 0 \} \) is connected.

1. If \( \phi \in H^1(G, {\mathbb Z}) \) then \( \phi \in \Sigma(G) \) if and only if there is a finitely generated subgroup \( H \subset G \) and an injective morphism \( f : H \to H \) such that \( G = H *_f \) and \( \phi \) is the induced character.
2. \( \phi \in \Sigma(G) \) if and only if \( \ker(\phi) \) is finitely generated.

For a 3–manifold group we have the following topological interpretation.

Theorem (Bieri–Neumann–Strebel + Perelman): Let \( M \) be a closed 3–manifold; then \( \phi \in \Sigma(\pi_1(M)) \) if and only if \( M \) is fibered and \( \phi \) is the induced character of the resulting HNN-extension. Moreover \( \phi \in \Sigma(\pi_1(M)) \) if and only if \( -\phi \in \Sigma(\pi_1(M)) \).

The Thurston norm

Let \( M \) be a compact 3–manifold. For a class \( \phi \in H^1(M; _ZZ) \) its Thurston norm \( \| \phi \|_T \) is defined as \( \min_S \chi_-(S) \) where \( S \) runs over all surfaces in \( M \) dual to \( \phi \) and \( \chi_-(S) = \sum_i \max(0, -\chi(S_i)) \) if \( S \) is the disjoint union of the connected surfaces \( S_i \).

Theorem (Thurston):

• \( \| \cdot \|_T \) extends to a semi-norm on \( H^1(M, {\mathbb R}) \).
• If \( M \) is hyperbolic then it is a norm.
• The unit ball is a polytope, and \( \Sigma(\pi_1(M)) \) is the cone over a union of maximal open faces of this polytope.
• If \( M \) is fibered withfiber \( \Sigma \) and induced character \( \phi \) then \( \| \phi \|_T = -\chi(\Sigma) \).

Universal \( L^2 \)-torsion

Let \( G \) have a finite, \( L^2 \)-acyclic \( K(G, 1) \), and in addition satisfy the Atiyah conjecture. Let \( {\mathbb Z} G \subset \mathcal DG \) be the division closure of \( {\mathbb Z} G \) in the algebra of affiliated operators (see Thomas Schick’s talk).

Let \( K_1^w(G) \) be the group generated by weak isomorphisms \( {\mathbb Z} G^n \to {\mathbb Z} G^n \) (maps which are invertible over \( \mathcal DG \)) and the same relations as in the usual \( K_1 \). Then Friedl–Lück define the universal \( L^2 \)-torsion \( \rho_w^{(2)}(G) \in K_1^w(G)\) as the classical Reidemeister torsion. Using the Dieudonné determinant this yields an invariant \( \det(\rho_u^{(2)}) \) in the abelianised group \( (\mathcal DG^\times) \).

Let \( G^{\mathrm{ab-f}} \) be the maximal free abelian quotient if \( G \) and \( K \) the kernel of \( G \to G^{\mathrm{ab-f}} \). Then \( \mathcal DG \) is the Ore localisation of the polynomial ring \( \mathcal DK[G^{\mathrm{ab-f}}] \), so we can write \( \det(\rho_u^{(2)}) = pq^{-1} \) for some \( p, q \in \mathcal DK[G^{\mathrm{ab-f}}] \).

Now for any \( p \in \mathcal DK[G^{\mathrm{ab-f}}] \) its support is a finite subset of \( G^{\mathrm{ab-f}} = H_1(G) \). Let \( \mathcal P(p) \) be the convex hull in \( H_1(G; {\mathbb R}) \) of this support.

For a polytope \( \mathcal P \subset H_1(G; {\mathbb R}) \) define the function \( \mathcal N(\mathcal P) : H^1(G; {\mathbb R}) \to [0, +\infty[ \) by :
\[
\mathcal N(\mathcal P)(\phi) = \sup_{a, b \in \mathcal P}(\phi(a) – \phi(b)).
\]
We get a canonical function on \( H^1(G; {\mathbb R}) \) by taking \( \mathcal N(\mathcal P(p)) – \mathcal N(\mathcal P(q)) \) where \( \det \rho_u^{(2)}(G) = pq^{-1} \).

Theorem (Friedl–Lück): If \( G = \pi_1(M) \) for \( M \) a 3–manifold then this function is equal to the Thurston norm.

Now this can be taken as a definition for the Thurston norm for an arbitrary group \( G \) which has a finiet \( K(G, 1) \) and which satisfies the conditions above (being \( L^2 \)-acyclic, satisfying the Atiyah conjecture and \( \mathrm{Wh}(G) = 0 \)).

Theorem (Linnell, Lück, Waldhausen): Any ascending HNN extension of a finitely generated free group satisfies these conditions.

So for this class of groups there is a well-defined function \( \|\cdot\|_T \) on \( H^1(G; {\mathbb R}) \).

Theorem (Funke–Kielak): In this setting \( \|\cdot\|_T \) is a semi-norm.

Comparison with the Alexander norm

For any finitely generated group \( G \) the Alexander polynomial \( \Delta_G \) is an element of the group ring \( {\mathbb Z} G^{\mathrm{ab-f}} \). Define the Alexander norm \( \|\cdot\|_A \) of \( G \) as the function \( \mathcal N(\mathcal P(\Delta_G)) \). Then McMullen proved that:

• If \( M \) is a 3–manifold with \( b_1(M) \ge 2 \) and \( G = \pi_1(M) \) then \( \|\cdot\|_A \le \|\cdot\|_T \);
• If in addition \( M \) is fibered with induced character \( \phi \) then \( \|\phi\|_A = \|\phi\|_T \).

(For non-fibered 3–manifolds the inequality can be strict as shown by Dunfield.)

Theorem (Funke–Kielak):If \( G = F_n * _f \) where either \( n = 2 \) or \( f \) is unipotent and polynomially growing, and \( b_1(G) \ge 2 \) then \( \|\cdot\|_A \le \|\cdot\|_T \) with equality on the BNS-invariant.

Theorem (Funke–Kielak):Under the same hypotheses the universal \( L^2 \)-torsion \( \rho_u^{(2)} \) determines the BNS-invariant.