## Notes of Thierry Coulbois’ talk

Botanic of Trees in Outer Space

Joint work with Arnaud Hilion

I am interested in particular examples of trees with isometric actions of a free group ${F}$. These are real trees, action is minimal (all orbits are dense), very small (all arc stabilisers are trivial). These represent points of Outer space (case of free actions) and its boundary.

1. Limit set

1.1. Definition

I am interested in a new notion of limit set: ${\Omega\subset \hat{T}:=\bar{T}\cup\partial T}$ is the set of points in the tree at which orbits accumulate in at least two different ways.

Definition 1 ${q\in\Omega}$ if there exist distinct points ${x}$, ${y\in\partial F}$ and sequences ${u_n}$, ${v_n \in F}$ converging respectively to ${x}$ and ${y}$ and such that ${u_n p}$ and ${v_n p}$ tend to ${q}$.

The boundary map ${\mathcal{Q}:\partial F \rightarrow \hat{T}}$, ${x\mapsto q}$ is continuous provided ${\hat{T}}$ is equipped with the weak observer topology (whose open sets are directions ${d\in\pi_0 (\hat{T}\setminus\{p\})}$.

1.2. Examples

1. Start with a once punctured torus. View it as the mapping torus of ${x\mapsto x+\alpha}$ on ${{\mathbb R}/{\mathbb Z}}$. It carries a measured foliation. Lift to universal covering space. There, the space of leaves is a metric tree with an isometric action of ${F_2}$. If ${\alpha\notin{\mathbb Q}}$, orbits are dense and ${\Omega=T}$.

2. The Levitt tree. Start with a band complex which is not a surface. E.g. more than two bands can be glued along the same interval. View this as the mapping torus of an interval translation mapping (more general that an interval exchange). Here, ${\Omega}$ is totally disconnected. Indeed, points of ${\Omega}$ are those which belong to bi-infinite leaves. Run Rips’ machine: erase the points which are endpoints of at most one band. Iterate. What remains is the relative limit set ${\Omega_0}$. It is a Cantor set. Then ${\Omega=F\Omega_0}$ has empty interior and is totally disconnected.

Theorem 2 (Coulbois, Hilion, Lustig) Every tree ${T}$ in Outer space and its boundary is transverse to the foliation of a band complex built on a compact subtree ${K}$ of ${\bar{T}}$.

Band complex is constructed as follows. Pick a basis of free group ${F}$. It restricts to a partial isometry of ${K}$. Glue a band ${dom(a)\times[0,1]}$ to ${K}$.

Here is our botanic. Trees split into 4 types,

1. Surface type trees: ${T\subset\Omega}$.
2. Levitt type trees: ${\Omega}$ is totally disconnected.
3. Geometric trees: ${K}$ is a finite tree.
4. Non geometric trees: see below.

2. Indices

To distinguish these types, we introduce numerical invariants. For simplicity, let us assume the ${F}$ action is free.

2.1. Geometric index

Definition 3 For ${p\in T}$, set

$\displaystyle \begin{array}{rcl} i_{geom}(p)=|\pi_0 (T\setminus\{p\}|-2. \end{array}$

Then define the geometric index of ${T}$ by

$\displaystyle \begin{array}{rcl} i_{geom}(T)=\sum_{[p]\in T/F}i_{geom}(p). \end{array}$

Theorem 4 (Gaboriau, Levitt) For all trees in Outer space of ${F_N}$ and its boundary, the geometric index satisfies

$\displaystyle \begin{array}{rcl} i_{geom}\leq 2N-2 \end{array}$

with equality iff ${T}$ is geometric.

2.2. ${Q}$-index

Again, assume the ${F_N}$-action is free.

Definition 5 For ${p\in \bar{T}}$, set

$\displaystyle \begin{array}{rcl} i_{Q}(p)=\max\{|Q^{-1}(p)|-2,0\}. \end{array}$

Then define the ${Q}$-index of ${T}$ by

$\displaystyle \begin{array}{rcl} i_{Q}(T)=\sum_{[p]\in T/F}i_{Q}(p). \end{array}$

Theorem 6 (Coulbois, Hilion) For all trees in the boundary of Outer space of ${F_N}$, the ${Q}$-index satisfies

$\displaystyle \begin{array}{rcl} i_{Q}\leq 2N-2 \end{array}$

with equality iff ${T}$ is of surface type.

Of surface type does not mean it arises from a surface, it is more general.

Example: Let ${\phi\in Out(F_N)}$ be iwip. Then the action on ${\phi}$ on the boundary of Outer Space has two fixed points which are trees with actions all of whose orbits are dense.

Proposition 7 If ${\phi}$ is iwip, both fixed points have the same ${Q}$-index.

Example 1 (Boshernitzian, Kornfeld) Let ${\phi}$ map ${a\mapsto b}$, ${b\mapsto caaa}$, ${c\mapsto caa}$. Then the fixed point ${T_{\phi^{-1}}}$ is our second example (Levitt tree), it is geometric, but ${i_Q (T)<2N-2}$. Whereas ${T_\phi}$ has ${i_Q (T_\phi)=2N-2}$, it is surface type. It has a compact heart ${K}$ of Hausdorff dimension ${>1}$ (${T}$ is incomplete and ${K}$ is contained in the metric completion). Furthermore, ${i_{geom}(T_\phi)<2N-2}$ so it is non geometric.