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 be a group, . Form the mapping torus . This is an extension of with kernel .
The group theoretic analogue of the question of identifying fibrations over the circle is
Question. Let be a group. When does an epimorphism arise from a mapping torus ?
An obvious necessary condition is that be finitely generated. It is known to be sufficient in the case of closed 3-manifolds.
2. Case of 3-manifolds
Let be a closed 3-manifold. Thurston has defined a semi-norm on with the following properties.
- The unit ball is a polyhedron with rational vertices.
- An epimorphism comes from a fibration (i.e. is finitely generated) if and only if belongs to the cone over certain open top-dimensional face.
Using Poincaré duality , Thurston’s norm on 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 is coherent (all finitely generated subgroups are finitely presented.
Bridson-Graner: has at worst quadratic Dehn function.
Gautero-Lustig: 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 on generators grows polynomially. This phenomenon does not arise for surface groups
has quadratic growth.
Theorem 1 (Cashen-Levitt) Let have polynomial growth. Let . There exist such that, given an epimorphism ,
- if for some , , then is not finitely generated.
- if for all , then is finitely generated, free, and its rank is
5. Connection with the Bieri-Neumann-Strebel invariant
Let be a finitely generated group. The Bieri-Neumann-Strebel invariant is a positively invariant subset of . Choose a Cayley graph. A homomorphism defines a map .
Examples. everything for . is empty for .
- is independant on the choice of .
- is open.
- If is rational, is finitely generated if and only if .
- everything if and only if the commutator subgroup is finitely generated.
Example. has only one map to , so is a line, is half of this line.
Example. Let be the following variant of Thompson’s group. is the group of piecewise linear homeos of with slopes in , and breaks at . Then 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.