论文标题
对粗组的邀请
An Invitation to Coarse Groups
论文作者
论文摘要
在这本专着中，我们为粗略的群体和粗糙作用的理论奠定了基础。粗组是粗空间类别中的组对象，可以将其视为套件，并具有满足组公理“均匀边界误差”的操作。在这项工作的第一部分中，我们发展了粗大的同态，商和亚组的理论，并证明了同构定理的粗糙版本是正确的。我们还启动了粗略动作的研究，并展示了它们与几何群体理论的基本观察如何相关。在第二部分中，我们探讨了专业主题的选择，例如对群体组的粗组结构的研究，一组粗大的自动形态和受控地图的空间。这里的主要目的是展示粗体群体的理论如何与经典主题联系起来。其中包括：数字理论；对小组的双重变量指标的研究；准态和稳定的换向器长度； $ {\ rm out}（f_n）$;拓扑团体行动。
In this monograph we lay the foundation for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms "up to uniformly bounded error". In the first part of this work, we develop the theory of coarse homomorphisms, quotients, and subgroups, and prove that coarse versions of the Isomorphism Theorems hold true. We also initiate the study of coarse actions and show how they relate to the fundamental observation of Geometric Group Theory. In the second part we explore a selection of specialized topics, such as the study of coarse group structures on set-groups, groups of coarse automorphisms and spaces of controlled maps. Here the main aim is to show how the theory of coarse groups connects with classical subjects. These include: number theory; the study of bi-invariant metrics on groups; quasimorphisms and stable commutator length; ${\rm Out}(F_n)$; topological group actions.
