论文标题
在球体复合物上的渐近翻译长度和广义的纤维锥上
On the asymptotic translation lengths on the sphere complexes and the generalized fibered cone
论文作者
论文摘要
在本文中，我们研究了在整个圆圈上纤维纤维的单个单粒体的渐近翻译长度。给定一个紧凑的映射圆环，我们在第一个共同体中定义了一个锥体，我们称之为广义的纤维锥体，并表明每个原始积分元件都会在圆上振动。此外，我们证明了普遍的纤维锥是弗里德锥的合理切片，该切片被定义为同源性方向的双重，是瑟斯顿纤维锥的类似物。 由于我们对广义纤维锥的描述，我们提供了广义的纤维锥的每个适当子单元，并具有均匀的上限，用于在适当的子观念中纤维的球体复合物上的单个单粒子长度。我们的上限纯粹是根据适当子键的维度。我们还推断出在Dowdall-Kapovich-Leininger作品中构建的有限图上的某些映射类别的渐近翻译长度的相似估计值，该图表以相关的自由切割复合物和自由因素复合物进行测量。 此外，作为我们的结果的应用，我们证明了磁盘复合体上$ g $ handlebody群体最小渐近翻译长度的渐近线为$ 1/g^2 $，与曲线复合体上的渐近差为相同。
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.
