** 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

**Theorem 1** * *

*
*
- 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 .

* *

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri PoincarÃ©, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/