## Notes of Gilbert Levitt’s lecture

On mapping tori of automorphisms of free groups

Work in progress with Chris Cashen, I do not understand the topic completely.

1. Mapping tori

Let ${F}$ be a group, ${\alpha\in Aut(F)}$. Form the mapping torus ${G_\alpha=}$. This is an extension of ${{\mathbb Z}}$ with kernel ${F}$.

The group theoretic analogue of the question of identifying fibrations over the circle is

Question. Let ${G}$ be a group. When does an epimorphism ${G\rightarrow{\mathbb Z}}$ arise from a mapping torus ?

An obvious necessary condition is that ${\mathrm{Ker}(\phi)}$ be finitely generated. It is known to be sufficient in the case of closed 3-manifolds.

2. Case of 3-manifolds

Let ${M}$ be a closed 3-manifold. Thurston has defined a semi-norm on ${H^1(M,{\mathbb R})}$ with the following properties.

1. The unit ball is a polyhedron with rational vertices.
2. An epimorphism ${\pi_1(M)\rightarrow{\mathbb Z}}$ comes from a fibration (i.e. ${\mathrm{Ker}(\phi)}$ is finitely generated) if and only if ${\phi}$ belongs to the cone over certain open top-dimensional face.

Using Poincaré duality ${H^1(M,{\mathbb R})\simeq H_2(M,{\mathbb R})}$, Thurston’s norm on ${H_2}$ attaches toe a homology class the minimal genus of an embedded surface representing it.

In other works, we understand mapping tori of automorphisms of surface groups. We would like to pass to free groups.

3. What is known about mapping tori of free group automorphisms

Feighn-Handel have proved that ${G_\alpha}$ is coherent (all finitely generated subgroups are finitely presented.

Bridson-Graner: ${G_\alpha}$ has at worst quadratic Dehn function.

Gautero-Lustig: ${G_\alpha}$ is relatively hyperbolic.

Chris Cashen has a plan to classify such groups up to quasiisometry, but this seems very hard.

Doudall-Kapovich-Leininger: attemps to extend Thurston theory to irreducible automorphisms.

4. Polynomial growth

I will focus on the opposite case, automorphisms of polynomial growth. This means that the length of iterates on ${\phi}$ on generators grows polynomially. This phenomenon does not arise for surface groups

Example 1

$\displaystyle \begin{array}{rcl} a\rightarrow a\quad b\rightarrow ba\quad c\rightarrow cb \end{array}$

Theorem 1 (Cashen-Levitt) Let ${\alpha\in Aut(F_n)}$ have polynomial growth. Let ${G_\alpha=F_b \times_\alpha {\mathbb Z}}$. There exist ${t_1,\ldots,t_n\in G_\alpha}$ such that, given an epimorphism ${\phi:G_\alpha\rightarrow {\mathbb Z}}$,

1. if for some ${i}$, ${\phi(t_i)=0}$, then ${\mathrm{Ker}(\phi)}$ is not finitely generated.
2. if ${\phi(t_i)\not=0}$ for all ${i}$, then ${\mathrm{Ker}(\phi)}$ is finitely generated, free, and its rank is

$\displaystyle \begin{array}{rcl} 1+\sum_{i=1}^n |\phi(t_i)|. \end{array}$

5. Connection with the Bieri-Neumann-Strebel invariant

Let ${G}$ be a finitely generated group. The Bieri-Neumann-Strebel invariant ${\Sigma^1}$ is a positively invariant subset of ${Hom(G,{\mathbb R})\setminus\{0\}}$. Choose a Cayley graph. A homomorphism ${G\rightarrow{\mathbb R}}$ defines a map ${\pi:Cay(G,S)\rightarrow{\mathbb R}}$.

Definition 2

$\displaystyle \begin{array}{rcl} \phi\in\Sigma^1 \Leftrightarrow \pi^{-1}([0,+\infty)) \textrm{ is connected}. \end{array}$

Examples. ${\Sigma^1=}$ everything for ${{\mathbb Z}^2}$. ${\Sigma^1}$ is empty for ${F_n}$.

Facts.

1. ${\Sigma^1}$ is independant on the choice of ${S}$.
2. ${\Sigma^1}$ is open.
3. If ${\phi}$ is rational, ${\mathrm{Ker}(\phi)}$ is finitely generated if and only if ${\phi\in\Sigma^1 \cup -\Sigma^1}$.
4. ${\Sigma^1=}$ everything if and only if the commutator subgroup ${[G,G]}$ is finitely generated.

Example. ${G=BS(1,2)}$ has only one map to ${{\mathbb R}}$, so ${Hom(G,{\mathbb R})}$ is a line, ${\Sigma^1}$ is half of this line.

Example. Let ${G}$ be the following variant of Thompson’s group. ${G}$ is the group of piecewise linear homeos of ${[0,1]}$ with slopes in ${2^{\mathbb Z} 3^{\mathbb Z}}$, and breaks at ${{\mathbb Z}[\frac{1}{6}]}$. Then ${\Sigma^1}$ is the complement of two irrational lines.

So we see that surface and free groups seem to behave rather nicely, compared to other groups. More generally, the BNS invariant can often be computed for amalgams.