论文标题
长度在映射类组和映射托里的简单卷上的功能
Length functions on mapping class groups and simplicial volumes of mapping tori
论文作者
论文摘要
令$ m $为封闭的定向歧管。我们在$ m $的映射类组$ \ mathrm {mcg}（m）$上介绍了两个称为填充卷的数值不变性，$ m $的$ \ mathrm {mcg}（m）$，这些定义是根据$ m $的奇异边界填充规范的定义，无论是真实的和积分的系数。我们表明，填充量是$ \ mathrm {mcg}（m）$上的长度函数，我们证明，映射类$ f $的真实填充量等于相应的映射torus $ e_f $的简单卷，而$ f $的积分填充体积并不小于稳定的$ e__f $ $ e__f $的稳定集成量。 我们讨论了填充量的几个消失和非变化结果。作为应用程序，我们表明，相对于其单肌的单曲映射Tori的双曲线体积不属于其单型，并且整体边界上的真实和整体填充规范通常是非bilipschitz等效的。
Let $M$ be a closed orientable manifold. We introduce two numerical invariants, called filling volumes, on the mapping class group $\mathrm{MCG}(M)$ of $M$, which are defined in terms of filling norms on the space of singular boundaries on $M$, both with real and with integral coefficients. We show that filling volumes are length functions on $\mathrm{MCG}(M)$, we prove that the real filling volume of a mapping class $f$ is equal to the simplicial volume of the corresponding mapping torus $E_f$, while the integral filling volume of $f$ is not smaller than the stable integral simplicial volume of $E_f$. We discuss several vanishing and non-vanishing results for the filling volumes. As applications, we show that the hyperbolic volume of $3$-dimensional mapping tori is not subadditive with respect to their monodromy, and that the real and the integral filling norms on integral boundaries are often non-biLipschitz equivalent.
