论文标题
与图相关的属尼尔曼群岛上的Anosov差异性
Anosov diffeomorphisms on infra-nilmanifolds associated to graphs
论文作者
论文摘要
封闭的riemannian歧管上的Anosov差异性是表现出均匀双曲行为的一种动力学系统。因此，对它们的特性进行了深入的研究，包括哪些空间允许这种差异性。据推测，任何承认Anosov差异的封闭流形都是同构的，即对尼尔曼群岛的同构，即由离散的异构体组的1个连接的nilpotent Lie组的紧凑型商。这种猜想激发了描述哪些Instra-Nilmanifolds承认Anosov差异的问题。 到目前为止，大多数研究都集中在尼尔曼福尔德的受限类别上，尼尔曼叶夫人是统一晶格的1个连接的尼尔替尼谎言组的商。例如，Dani和Mainkar研究了与图形相关的Nilmanifolds的这个问题，这些尼尔曼福尔德构成了在自由尼尔尼氏谎言基团上模拟的尼尔曼群岛的自然概括。本文将其工作进一步推广到与图形相关的完整类的基础类别，从而导致仅取决于自律组在定义图上的诱导作用，从而导致了必要和充分的条件。作为一种应用，我们由循环自动群体构建了基于nilmanifolds的家族，该家族承认Anosov差异性群体从简单图上的忠实行动开始。
Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore their properties are intensively studied, including which spaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold, i.e. a compact quotient of a 1-connected nilpotent Lie group by a discrete group of isometries. This conjecture motivates the problem of describing which infra-nilmanifolds admit an Anosov diffeomorphism. So far, most research was focused on the restricted class of nilmanifolds, which are quotients of 1-connected nilpotent Lie groups by uniform lattices. For example, Dani and Mainkar studied this question for the nilmanifolds associated to graphs, which form the natural generalization of nilmanifolds modeled on free nilpotent Lie groups. This paper further generalizes their work to the full class of infra-nilmanifolds associated to graphs, leading to a necessary and sufficient condition depending only on the induced action of the holonomy group on the defining graph. As an application, we construct families of infra-nilmanifolds with cyclic holonomy groups admitting an Anosov diffeomorphism, starting from faithful actions of the holonomy group on simple graphs.
