循环群的McMullen多项式与Lipschitz流
McMullen polynomials and Lipschitz flows for free-by-cyclic groups
摘要：
考虑一个群G和一个满射u_0：\to，它使G分裂为一个半直积ker（u_0）\Z，其中ker（u_0）是一个可生成的自由群，φ \in（ker（u_0））可由一个扩展的不可约训练轨道映射表示。基于我们早期的工作[Dynamics on free-by-循环群，arXiv：1301.7739]，其中我们实现了G为π_1（X）的Eilenberg-Maclane 2-复形X配备了半流流形，并受到纤维双曲3-流形的McMullen的Teichmüller多项式的启发，我们构造了（X，X）的多项式不变量\m并研究其性质。具体地，\m确定H ^1（G; \R）中的凸多面体锥\C_X，凸实解析函数\H：\C_X \to \R，并且专门针对每个积分u \in \C_X给出积分劳伦多项式\m_u（n）。我们证明了\C_X等于（X，X）的“截面锥”（所有上同调类的凸包对偶于X的截面），并且对于每个具有第一返回映射f_u的（相容）截面Θ_u：Θ_PDMATHABMM Θ_u，特殊化\m_u（X）编码f_u的转移矩阵的特征多项式。更一般地说，对于每个类u \in \C_X，存在一个测地度量d_u和一个横截于X的余维1叶理Ω_u，使得在重新参数化后，流ε ^u_s通过局部e^{s \H（u）}-同态将Ω_u的叶映射到叶。除其他事项外，我们还证明了\C_X是等于（锥）的组成部分的BNS不变包含u_0和每个原始积分\in \C_X诱导分裂G作为一个上升的HNN-扩展在有限秩自由群沿内射自同态u。对于任何这样的分裂，我们证明了ε_u的拉伸因子正好由e^{ \H（u）}给出。特别地，我们看到\C_X和\H仅依赖于群G和满态u_0。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
