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

Abstract

Consider a group $G$ and an epimorphism $u_0:G\to \mathbb{Z}$ inducing a splitting of $G$ as a semidirect product ker$(u_0){⋊}_{\phi }\mathbb{Z}$ with ker$(u_0)$ a finitely generated free group and $\phi \in$ Out (ker$(u_0)$) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which we realized $G$ as ${\pi }_1(X)$ for an Eilenberg–Maclane 2-complex $X$ equipped with a semiflow $\psi$, and inspired by McMullen's Teichmüller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant $\mathfrak{m}\in \mathbb{Z}[H_1(G;\mathbb{Z})/$torsion] for $(X,\psi )$ and investigate its properties.

Specifically, $\mathfrak{m}$ determines a convex polyhedral cone ${\mathcal{C}}_X\subset H^1(G;\mathbb{R})$, a convex, real-analytic function $\mathfrak{H}:{\mathcal{C}}_X\to \mathbb{R}$, and specializes to give an integral Laurent polynomial ${\mathfrak{m}}_u(\zeta )$ for each integral $u\in {\mathcal{C}}_X$. We show that ${\mathcal{C}}_X$ 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 ${\Theta }_u\subset X$ with first return map $f_u:{\Theta }_u\to {\Theta }_u$, the specialization ${\mathfrak{m}}_u(\zeta )$ encodes the characteristic polynomial of the transition matrix of $f_u$. More generally, for every class $u\in {\mathcal{C}}_X$ there exists a geodesic metric $d_u$ and a codimension-1 foliation ${\Omega }_u$ of $X$ defined by a "closed 1-form" representing $u$ transverse to $\psi$ so that after reparametrizing the flow ${\psi }_s^u$ maps leaves of ${\Omega }_u$ to leaves via a local $e^{s\mathfrak{H}(u)}$-homothety.

Among other things, we additionally prove that ${\mathcal{C}}_X$ is equal to (the cone over) the component of the BNS-invariant $\Sigma (G)$ containing $u_0$ and, consequently, that each primitive integral $u\in {\mathcal{C}}_X$ induces a splitting of $G$ as an ascending HNN-extension $G=Q_u{\ast }_{{\varphi }_u}$ with $Q_u$ a finite-rank free group and ${\varphi }_u:Q_u\to Q_u$ injective. For any such splitting, we show that the stretch factor of ${\varphi }_u$ is exactly given by $e^{\mathfrak{H}(u)}$. In particular, we see that ${\mathcal{C}}_X$ and $\mathfrak{H}$ depend only on the group $G$ and epimorphism $u_0$.