McMullen polynomials and Lipschitz flows for free-by-cyclic groups

Abstract

Consider a group G and an epimorphism u_0:G\to\Z inducing a splitting of G as a semidirect product ker(u_0)\rtimes_φ\Z with ker(u_0) a finitely generated free group and φ\in Out(ker(u_0)) representable by an expanding irreducible train track map. Building on our earlier work [Dynamics on free-by-cyclic groups, arXiv:1301.7739], in which we we realized G as π_1(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 for (X,ψ) and investigate its properties. Specifically, \m determines a convex polyhedral cone \C_X in H^1(G;\R), a convex, real-analytic function \H:\C_X\to\R, and specializes to give an integral Laurent polynomial \m_u(ζ) for each integral u\in\C_X. We show that \C_X 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 Θ_u with first return map f_u:Θ_u\toΘ_u, the specialization \m_u(ζ) encodes the characteristic polynomial of the transition matrix of f_u. More generally, for every class u\in\C_X there exists a geodesic metric d_u and a codimension-1 foliation Ω_u of X transverse to ψso that after reparametrizing the flow ψ^u_s maps leaves of Ω_u to leaves via a local e^{s\H(u)}-homothety. Among other things, we additionally prove that \C_X is equal to (the cone over) the component of the BNS-invariant containing u_0 and that each primitive integral u\in\C_X induces a splitting of G as an ascending HNN-extension over a finite-rank free group along an injective endomorphism ϕ_u. For any such splitting, we show that the stretch factor of ϕ_u is exactly given by e^{\H(u)}. In particular, we see that \C_X and \H depend only on the group G and epimorphism u_0.