Free-by-cyclic groups and trees
Joint work with S. Dowdall and I. Kapovich.
The Bieri-Neumann-Strebel invariant is an open subset of , it is the set of such that is surjective on . Here, is the torsion free abelian cover of and is an equivariant map representing .
If is free-by-cyclic, one can refine
Geoghegan-Mihalik-Sapir-Wise show that for every , is locally free and there exists an outer automorphism and a finitely generated subgroup such that . In particular, if , then one can take .
From now on, we assume that is atoroidal and fully irreducible. Then is hyperbolic, and there exists an expanding irreducible train track representative (Bestvina-Handel). Let be the mapping torus. It carries the suspension of , which is a one-sided flow (action of semi-group ). The representative of integral cohomology class factors to a map . Let be the subset of cohomology classes such that the representative can be chosen to be increasing along the flow. Then
- is a component of . It is a rational polyhedral cone.
- For , inverse images of points are cross-sections of the flow. The first return map is an expanding irreducible train track representative of , with .
Stretch factors form a nice function on .
Theorem 2 (Algom-Kfir-Hironaka-Rafi) There exists an -analytic, convex function such that for all such that for al and ,
- If , then .