## Notes of Yael Algom-Kfir’s Cambridge lecture 21-06-2017

Boundary of hyperbolic free-by-cyclic groups

Joint work with Hilion and Starck.

Let ${\phi\in Out(F_n)}$. We are interested in the HNN extension ${G=\langle F_n,t|txt^{-1}=\phi(x)\rangle}$.

If ${\phi}$ is atoroidal (i.e. no conjugacy class is fixed), then ${G}$ is Gromov hyperbolic, its boundary is compact and metrizable. ${\partial G}$ is connected if ${G}$ does not virtually split over a finite subgroup. We know that ${cd(G)=2}$, hence dim${(G)=1}$ (Bestvina-Mess). ${\partial G}$ is locally connected. If furthermore ${\phi}$ does not preserve a free splitting, then ${G}$ does not split over a virtually cyclic subgroup, hence ${\partial G}$ has no local cut points. A topological space with these properties is

• either planar, hence a Sierpinski carpet,
• or no open subset of is planar, hence a Menger curve.

Theorem 1 (Algom-Kfir-Hilion-Starck) If ${\phi}$ is atoroidal, then there is an embedding of ${K_{3,3}}$ in ${\partial G}$.

Thus ${\partial G}$ is always a Menger curve.

1. Outer space

Culler-Vogtman’s Outer space ${CV_n}$ is the space of free isometric actions of ${F_n}$ on metric simplicial trees, up to homothety. Its compactification ${\overline{CV}_n}$ is the space of isometric actions of ${F_n}$ on metric ${{\mathbb R}}$-trees with very small stabilizers, up to homothety.

${Out(F_n)}$ acts on this space. Each atoroidal ${\phi}$ acts by a North-South dynamics, with a repellor and and an attractor trees ${T^\pm}$ with ${F_n}$ actions. These trees have dense orbits and are indecomposable. An isometric action induces a continuous equivariant map ${Q:\partial F_n\rightarrow T\cup \partial T=\hat T}$ (Levitt-Lustig). Define the associated lamination

$\displaystyle \begin{array}{rcl} L(T)=\{(X,Y)\in\partial^2 F_n\,;\,Q(X)=Q(Y)\}. \end{array}$

It is closed and flip-invariant. Hence maps ${Q_\pm}$ and spaces ${L_\pm}$.

Mitra proved that the injection ${F_n\rightarrow G}$ extends continuously to ${i_{CT}:\partial F_n\rightarrow\partial G}$. Kapovich-Lustig showed that ${i_{CT}}$ factors via ${Q_\pm}$ and maps

$\displaystyle \begin{array}{rcl} R_\pm:\hat T_\pm\rightarrow\partial G. \end{array}$

They are injective on ${T_\pm}$, map them to disjoint subsets, and satisfy ${R_+(X)=R_+(Y)}$ iff ${X=Q_+(\xi)}$ and ${Y=Q_+(\eta)}$ with ${(\xi,\eta)\in L(T_-)}$.

Using arcs in ${\partial G}$, we manage to produce a complete bi-partite graph in ${\partial G}$, but this requires some control on combinatorics of arcs.

I recall the notion of Whitehead graphs (Coulbois-Hilion-Reynolds).

Given ${L(T_-)}$ and a branch point ${v\in T_-}$, define

$\displaystyle \begin{array}{rcl} W_{T_-}(v)=\{\textrm{edges corresponding to }(\xi,\eta)\in L(T_-)'\}. \end{array}$

We need a refined notion of directional Whitehead graph. A direction is an edge of ${T_+}$ emanating from ${v}$.

3. Question

What is the conformal structure on ${\partial G}$ like ?