**Quasiisometries of mapping tori of free group automorphisms**

Joint work with Natalia Macura

**1. The problem **

Definition 1Let . The corresponding mapping torus group is the semi-direct product . It depends only on the class .

Goal: classify these groups up to quasiisometry.

Note: if and have conjugate powers, then the mapping tori are commensurable.

Say that is polynomial growth of degree if for every , has polynomial growth of degree at most . This degree is a quasiisometry invariant of the mapping torus (Macura).

We shall stick to the polynomial case (even linear), since the exponential case is much different.

**2. General outline **

In order to show that two groups are quasiisometric,

- Find a geometric (i.e. quasiisometry invariant) decomposition.
- Check QI types of pieces.
- Check that pieces fit together in the right way.

** 2.1. Geometric decomposition **

This is what relative train tracks are about.

Theorem 2 (Bestvina, Feighn, Handel)If grows linearly, there exist

- a graph with fundamental group .
- a homotopy equivalence realizing on fundamental group.
- a filtration such that -invariant edges, and, at each step, one edge is added, and where is a Nielsen loop in .
- and some bounded cancellation condition on suffixes.

For instance, consider automorphisms of the form , , . Call a suffix. To simplify matters, I will assume that all suffixes are in .

In the example, one sees an HNN extension of direct product .

**Claim**: When is sufficiently complicated (to be made precise later), this is the JSJ decomposition of the mapping torus.

** 2.2. QI types of pieces **

In the example, only one piece, easy to understand.

** 2.3. Pieces fit together in the right way **

Above the Bass-Serre tree of the HNN extension, we have the Bass-Serre complex, which maps to the mapping torus. There is a line pattern on that complex, that records how edges glue on.

A quasiisometry between mapping tori must restrict to a quasiisometry of that respects line partitions.

**Question**: Given word , look at coarse equivalence classes of -lines, i.e. sets of the form . I.e., is there a quasiisometry such that induces a bijection between line partitions and ? Similar question for multicurves.

** 2.4. Examples **

Use Whitehead’s algorithm.

**Observation**: Let denote the Cayley tree of , its ideal boundary. Each line of the line pattern has endpoints in . Every line pattern preserving quasiisometry of extends to a homeomorphism of that preserves pairs . It descends to a homeomorphism of the quotient of by equivalence relation .

Construct the Whitehead graph with vertices , . Add edges from to for each occurrence of in . Whitehead’s algorithm consists in reducing the number of edges by using special automorphisms. This leads easily to following theorem.

Theorem 3The following are equivalent.

- Any Whitehead graph is not connected.
- Every minimal Whitehead graph is not connected.
- is not connected.
- splits freely relative to (i.e. F is a free product with in one of the factors).

The following theorem extends results of Otal and is analogous to a theorem of Bowditch.

Theorem 4If the decomposition space is connected and not a circle, the following are equivalent.

- has cut points or cut pairs.
- splits cyclically relative to .

A theorem of Otal tells when is a circle.

Theorem 5 (Otal)The following are equivalent.

- is a circle.
- bounds a surface.
- Any minimal Whitehead graph is a circle.

Theorem 6 (Cashen, Macura)If the decomposition space is connected with no cut points and no cut pairs, then there exist a cube complex such that

- A quasiisometry .
- An isometric action of on which, restricted to , is cocompact.

Proof is an extension of the proof of Stallings’ theorem.

**Example**: . -point cut sets occur. The cube complex encodes the combinatorics if these -point cut sets.