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

### Spencer Dowdall

University of Illinois at Urbana-Champaign, USA### Ilya Kapovich

University of Illinois at Urbana-Champaign, USA### Christopher J. Leininger

University of Illinois at Urbana-Champaign, USA

## Abstract

Consider a group $G$ and an epimorphism $u_{0}:G→Z$ inducing a splitting of $G$ as a semidirect product ker$(u_{0})⋊_{φ}Z$ with ker$(u_{0})$ a finitely generated free group and $φ∈$ Out (ker$(u_{0})$) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which 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 $m∈Z[H_{1}(G;Z)/$torsion] for $(X,ψ)$ and investigate its properties.

Specifically, $m$ determines a convex polyhedral cone $C_{X}⊂H_{1}(G;R)$, a convex, real-analytic function $H:C_{X}→R$, and *specializes* to give an integral Laurent polynomial $m_{u}(ζ)$ for each integral $u∈C_{X}$. 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 $Θ_{u}⊂X$ with first return map $f_{u}:Θ_{u}→Θ_{u}$, the specialization $m_{u}(ζ)$ encodes the characteristic polynomial of the transition matrix of $f_{u}$. More generally, for *every* class $u∈C_{X}$ there exists a geodesic metric $d_{u}$ and a codimension-1 foliation $Ω_{u}$ of $X$ defined by a "closed 1-form" representing $u$ transverse to $ψ$ so that after reparametrizing the flow $ψ_{s}$ maps leaves of $Ω_{u}$ to leaves via a local $e_{sH(u)}$-homothety.

Among other things, we additionally prove that $C_{X}$ is equal to (the cone over) the component of the BNS-invariant $Σ(G)$ containing $u_{0}$ and, consequently, that each primitive integral $u∈C_{X}$ induces a splitting of $G$ as an ascending HNN-extension $G=Q_{u}∗_{ϕ_{u}}$ with $Q_{u}$ a finite-rank free group and $ϕ_{u}:Q_{u}→Q_{u}$ injective. 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 $u_{0}$.

## Cite this article

Spencer Dowdall, Ilya Kapovich, Christopher J. Leininger, McMullen polynomials and Lipschitz flows for free-by-cyclic groups. J. Eur. Math. Soc. 19 (2017), no. 11, pp. 3253–3353

DOI 10.4171/JEMS/739