论文标题
对角线类型及其基本组的纤维平坦歧管
Fibering flat manifolds of diagonal type and their fundamental groups
论文作者
论文摘要
据说，如果其自律组$ g $的标准表示是对角线的，则据说$ n $维的封闭平面歧管是对角类型的。 $ n $维数的对角线类型是这种歧管的基本组。我们将$ g $的对角线vasquez不变式介绍为最小整数$ n_d（g）$，使得对角线类型的每种平坦多种流形带有固体$ g $纤维在尺寸的平坦多数$ n_d（g）$的平坦歧管上。使用对角线类型的Bieberbach组的组合描述，我们给出了这种不变的上限和下限。我们表明，当$ g $的排名低时，下限是准确的。我们将其应用于分析对角线类型的Bieberbach组的扩散特性。这导致对对角线类型的Bieberbach组进行了完整的分类，并具有Klein四组全能情况，并应用于Kaplansky的单位猜想。
An $n$-dimensional closed flat manifold is said to be of diagonal type if the standard representation of its holonomy group $G$ is diagonal. An $n$-dimensional Bieberbach group of diagonal type is the fundamental group of such a manifold. We introduce the diagonal Vasquez invariant of $G$ as the least integer $n_d(G)$ such that every flat manifold of diagonal type with holonomy $G$ fibers over a flat manifold of dimension at most $n_d(G)$ with flat torus fibers. Using a combinatorial description of Bieberbach groups of diagonal type, we give both upper and lower bounds for this invariant. We show that the lower bounds are exact when $G$ has low rank. We apply this to analyse diffuseness properties of Bieberbach groups of diagonal type. This leads to a complete classification of Bieberbach groups of diagonal type with Klein four-group holonomy and to an application to Kaplansky's Unit Conjecture.
