JournalsjemsVol. 19, No. 11pp. 3253–3353

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

Consider a group GG and an epimorphism u0 ⁣:GZu_0\colon G\to \mathbb Z inducing a splitting of GG as a semidirect product ker(u0)φZ(u_0)\rtimes_\varphi \mathbb Z with ker(u0)(u_0) a finitely generated free group and φ\varphi\in Out (ker(u0)(u_0)) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which we realized GG as π1(X)\pi_1(X) for an Eilenberg–Maclane 2-complex XX equipped with a semiflow ψ\psi, and inspired by McMullen's Teichmüller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant mZ[H1(G;Z)/\mathfrak m \in \mathbb Z[H_1(G;\mathbb Z)/torsion] for (X,ψ)(X,\psi) and investigate its properties.

Specifically, m\mathfrak m determines a convex polyhedral cone CXH1(G;R)\mathcal C_X\subset H^1(G;\mathbb R), a convex, real-analytic function H ⁣:CXR\mathfrak H\colon \mathcal C_X\to \mathbb R, and specializes to give an integral Laurent polynomial mu(ζ)\mathfrak m_u(\zeta) for each integral uCXu\in \mathcal C_X. We show that CX\mathcal C_X is equal to the "cone of sections" of (X,ψ)(X,\psi) (the convex hull of all cohomology classes dual to sections of of ψ\psi), and that for each (compatible) cross section ΘuX\Theta_u\subset X with first return map fu ⁣:ΘuΘuf_u\colon \Theta_u\to \Theta_u, the specialization mu(ζ)\mathfrak m_u(\zeta) encodes the characteristic polynomial of the transition matrix of fuf_u. More generally, for every class uCXu\in \mathcal C_X there exists a geodesic metric dud_u and a codimension-1 foliation Ωu\Omega_u of XX defined by a "closed 1-form" representing uu transverse to ψ\psi so that after reparametrizing the flow ψsu\psi^u_{s} maps leaves of Ωu\Omega_u to leaves via a local esH(u)e^{s\mathfrak H(u)}-homothety.

Among other things, we additionally prove that CX\mathcal C_X is equal to (the cone over) the component of the BNS-invariant Σ(G)\Sigma(G) containing u0u_0 and, consequently, that each primitive integral uCXu\in \mathcal C_X induces a splitting of GG as an ascending HNN-extension G=QuϕuG = Q_u\ast_{\phi_u} with QuQ_u a finite-rank free group and ϕu ⁣:QuQu\phi_u\colon Q_u\to Q_u injective. For any such splitting, we show that the stretch factor of ϕu\phi_u is exactly given by eH(u)e^{\mathfrak H(u)}. In particular, we see that CX\mathcal C_X and H\mathfrak H depend only on the group GG and epimorphism u0u_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