Mcmullen Polynomials and Lipschitz Flows for Free-by-cyclic Groups

Consider a group G and an epimorphism u0 : G → Z inducing a splitting of G as a semidirect product ker(u0)oφ Z with ker(u0) a finitely generated free group and φ ∈ Out(ker(u0)) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which we realized G as π1(X) for an Eilenberg-Maclane 2–complex X equipped with a semiflow ψ, and inspired by McMullen’s Teichmüller polynomial for fibered hyperbolic 3–manifolds, we construct a polynomial invariant m ∈ Z[H1(G;Z)/torsion] for (X,ψ) and investigate its properties. Specifically, m determines a convex polyhedral cone CX ⊂ H1(G;R), a convex, real-analytic function H : CX → R, and specializes to give an integral Laurent polynomial mu(ζ) for each integral u ∈ CX . We show that CX is equal to the “cone of sections” of (X,ψ) (the convex hull of all cohomology classes dual to sections of of ψ), and that for each (compatible) cross section Θu ⊂ X with first return map fu : Θu → Θu, the specialization mu(ζ) encodes the characteristic polynomial of the transition matrix of fu. More generally, for every class u ∈ CX there exists a geodesic metric du and a codimension–1 foliation Ωu of X defined by a “closed 1–form” representing u transverse to ψ so that after reparametrizing the flow ψu s maps leaves of Ωu to leaves via a local esH(u)–homothety. Among other things, we additionally prove that CX is equal to (the cone over) the component of the BNSinvariant Σ(G) containing u0 and, consequently, that each primitive integral u ∈ CX induces a splitting of G as an ascending HNN-extension G = Qu∗φu with Qu a finite-rank free group and φu : Qu → Qu injective. For any such splitting, we show that the stretch factor of φu is exactly given by eH(u). In particular, we see that CX and H depend only on the group G and epimorphism u0.