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_\varphi\Z with ker(u_0) a finitely generated free group and \varphi\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 \pi_1(X) for an Eilenberg-Maclane 2-complex X equipped with a semiflow \psi, and inspired by McMullen's Teichm\"uller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant \m for (X,\psi) 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(\zeta) for each integral u\in\C_X. We show that \C_X is equal to the "cone of sections" of (X,\psi) (the convex hull of all cohomology classes dual to sections of of \psi), and that for each (compatible) cross section \Theta_u with first return map f_u:\Theta_u\to\Theta_u, the specialization \m_u(\zeta) 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 \Omega_u of X transverse to \psi so that after reparametrizing the flow \psi^u_s maps leaves of \Omega_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 \phi_u. For any such splitting, we show that the stretch factor of \phi_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.