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.

2. Directional Whitehead graphs

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 ?


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