## Notes of Chris Cashen’s lecture

Quasiisometries of mapping tori of free group automorphisms

Joint work with Natalia Macura

1. The problem

Definition 1 Let ${\alpha\in Aut(F)}$. The corresponding mapping torus group is the semi-direct product ${F\times_{\alpha}{\mathbb Z}}$. It depends only on the class ${[\alpha]\in Out(F)}$.

Goal: classify these groups up to quasiisometry.

Note: if ${\alpha}$ and ${\beta}$ have conjugate powers, then the mapping tori are commensurable.

Say that ${\alpha}$ is polynomial growth of degree ${d}$ if for every ${f\in F}$, ${|\alpha^k (f)|}$ has polynomial growth of degree at most ${d}$. 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,

1. Find a geometric (i.e. quasiisometry invariant) decomposition.
2. Check QI types of pieces.
3. 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 ${[\alpha]}$ grows linearly, there exist

• a graph ${\Gamma}$ with fundamental group ${F}$.
• a homotopy equivalence ${\alpha}$ realizing ${[\alpha]}$ on fundamental group.
• a filtration ${\Gamma_0 \subset \Gamma_1 \subset\cdots\subset\Gamma_k =\Gamma}$ such that ${\Gamma_0 = \{\alpha}$-invariant edges${\}}$, and, at each step, one edge ${e}$ is added, and ${\alpha(e)=e\gamma}$ where ${\gamma}$ is a Nielsen loop in ${\Gamma_{i-1}}$.
• and some bounded cancellation condition on suffixes.

For instance, consider automorphisms of the form ${a\mapsto a}$, ${b\mapsto a}$, ${c\mapsto cw}$. Call ${w}$ a suffix. To simplify matters, I will assume that all suffixes are in ${\Gamma_0}$.

In the example, one sees an HNN extension of direct product ${\langle a,b\rangle\times{\mathbb Z}}$.

Claim: When ${w}$ 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 ${F\times{\mathbb Z}}$ that respects line partitions.

Question: Given word ${w\in F}$, look at coarse equivalence classes of ${w}$-lines, i.e. sets of the form ${\{uw^m\,;\,u\in F\}}$. I.e., is there a quasiisometry ${\phi:F\rightarrow F'}$ such that ${\phi}$ induces a bijection between line partitions ${\mathcal{L}}$ and ${\mathcal{L'}}$ ? Similar question for multicurves.

2.4. Examples

Observation: Let ${T}$ denote the Cayley tree of ${F}$, ${\partial T}$ its ideal boundary. Each line ${\ell\in\mathcal{L}}$ of the line pattern has endpoints ${\ell^{\pm}}$ in ${\partial T}$. Every line pattern preserving quasiisometry of ${F}$ extends to a homeomorphism of ${\partial T}$ that preserves pairs ${\{\ell^{+},\ell^{-}\}}$. It descends to a homeomorphism of the quotient ${\mathcal{D}}$ of ${\partial T}$ by equivalence relation ${\ell^{+}\sim\ell^{-}}$.

Construct the Whitehead graph with vertices ${x_i}$, ${\bar{x}_i}$. Add edges from ${\bar{x}_i}$ to ${x_j}$ for each occurrence of ${x_i x_j}$ in ${w}$. Whitehead’s algorithm consists in reducing the number of edges by using special automorphisms. This leads easily to following theorem.

Theorem 3 The following are equivalent.

1. Any Whitehead graph is not connected.
2. Every minimal Whitehead graph is not connected.
3. ${\mathcal{D}}$ is not connected.
4. ${F}$ splits freely relative to ${w}$ (i.e. F is a free product with ${w}$ in one of the factors).

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

Theorem 4 If the decomposition space ${\mathcal{D}}$ is connected and not a circle, the following are equivalent.

1. ${\mathcal{D}}$ has cut points or cut pairs.
2. ${F}$ splits cyclically relative to ${w}$.

A theorem of Otal tells when ${\mathcal{D}}$ is a circle.

Theorem 5 (Otal) The following are equivalent.

1. ${\mathcal{D}}$ is a circle.
2. ${w}$ bounds a surface.
3. Any minimal Whitehead graph is a circle.

Theorem 6 (Cashen, Macura) If the decomposition space ${\mathcal{D}}$ is connected with no cut points and no cut pairs, then there exist a ${CAT(0)}$ cube complex ${X}$ such that

1. A quasiisometry ${T\rightarrow X}$.
2. An isometric action of ${QI(F,\mathcal{L})}$ on ${X}$ which, restricted to ${F}$, is cocompact.

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

Example: ${<=a^2 b^2 \bar{a}b}$. ${3}$-point cut sets occur. The cube complex encodes the combinatorics if these ${3}$-point cut sets.