论文标题
Metabelian团体的估值，完成和双曲线行动
Valuations, completions, and hyperbolic actions of metabelian groups
论文作者
论文摘要
双曲线度量空间的行动是研究组的重要工具，因此很难尝试对固定组的所有此类动作进行分类。在本文中，我们在双曲几何形状和交换代数之间建立了牢固的联系，以便将许多Metabelian群体的双曲线作用分类为粗糙的等效性。特别是，我们将此分类问题转变为在某些环的完成中对理想进行分类的问题，并计算矩阵的不变子空间。我们使用此框架来对许多与扩展整数矩阵相关的阿伯利亚人组的双曲线作用进行分类。每个这样的动作都等同于在树上或heintze群体上的动作（经典研究的一类负弯曲的谎言组）。我们的调查结合了数字系统，正式功率系列环，完成和估值中的分解。
Actions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian-by-cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations.
