关于球面复形上的渐近平移长度与广义纤维锥
On the asymptotic translation lengths on the sphere complexes and the generalized fibered cone
AI总结：
本文研究圆周纤维化流形的幺模映射在球面复形上的渐近平移长度，定义广义纤维锥并证明其为Fried锥的有理切片，给出其真子锥上渐近平移长度的统一上界，并应用于柄体群在圆盘复形上的最小渐近平移长度。
摘要：
本文研究了圆周上纤维化流形的幺模映射在球面复形上的渐近平移长度。给定一个紧致映射环面，我们在第一上同调中定义了一个锥，称之为广义纤维锥，并证明了每个本原整元素都给出一个圆周上的纤维化。此外，我们证明了广义纤维锥是Fried锥的一个有理切片，其中Fried锥被定义为同调方向的对偶，是Thurston纤维锥的一个类比。\n 作为我们对广义纤维锥描述的一个结果，我们为广义纤维锥的每个真子锥提供了该真子锥中纤维的幺模映射在球面复形上的渐近平移长度的统一上界。我们的上界仅取决于真子锥的维数。我们还推导了在Dowdall--Kapovich--Leininger的工作中构造的有限图上的某些映射类的渐近平移长度的类似估计，这些估计在相关的自由分裂复形和自由因子复形上测量。\n 此外，作为我们结果的一个应用，我们证明了亏格$g$的柄体群在圆盘复形上的最小渐近平移长度的渐近线是$1/g^2$，与曲线复形上的相同。
英文摘要：
In this paper, we study the asymptotic translation lengths on the sphere complexes of monodromies of a manifold fibered over the circle. Given a compact mapping torus, we define a cone in the first cohomology which we call the generalized fibered cone, and show that every primitive integral element gives a fibration over the circle. Moreover, we prove that the generalized fibered cone is a rational slice of Fried's cone, which is defined as the dual of homological directions, an analogue of Thurston's fibered cone. As a consequence of our description of the generalized fibered cone, we provide each proper subcone of the generalized fibered cone with a uniform upper bound for asymptotic translation lengths of monodromies on sphere complexes of fibers in the proper subcone. Our upper bound is purely in terms of the dimension of the proper subcone. We also deduce similar estimates for asymptotic translation lengths of some mapping classes on finite graphs constructed in the works of Dowdall--Kapovich--Leininger, measured on associated free-splitting complexes and free-factor complexes. Moreover, as an application of our result, we prove that the asymptote for the minimal asymptotic translation length of the genus $g$ handlebody group on the disk complex is $1/g^2$, the same as the one on the curve complex.
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