Boundary of hyperbolic free-by-cyclic groups
Joint work with Hilion and Starck.
Let . We are interested in the HNN extension .
If is atoroidal (i.e. no conjugacy class is fixed), then is Gromov hyperbolic, its boundary is compact and metrizable. is connected if does not virtually split over a finite subgroup. We know that , hence dim (Bestvina-Mess). is locally connected. If furthermore does not preserve a free splitting, then does not split over a virtually cyclic subgroup, hence 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 is atoroidal, then there is an embedding of in .
Thus is always a Menger curve.
1. Outer space
Culler-Vogtman’s Outer space is the space of free isometric actions of on metric simplicial trees, up to homothety. Its compactification is the space of isometric actions of on metric -trees with very small stabilizers, up to homothety.
acts on this space. Each atoroidal acts by a North-South dynamics, with a repellor and and an attractor trees with actions. These trees have dense orbits and are indecomposable. An isometric action induces a continuous equivariant map (Levitt-Lustig). Define the associated lamination
It is closed and flip-invariant. Hence maps and spaces .
Mitra proved that the injection extends continuously to . Kapovich-Lustig showed that factors via and maps
They are injective on , map them to disjoint subsets, and satisfy iff and with .
Using arcs in , we manage to produce a complete bi-partite graph in , but this requires some control on combinatorics of arcs.
2. Directional Whitehead graphs
I recall the notion of Whitehead graphs (Coulbois-Hilion-Reynolds).
Given and a branch point , define
We need a refined notion of directional Whitehead graph. A direction is an edge of emanating from .
What is the conformal structure on like ?