几何近似群论的基础
Foundations of geometric approximate group theory
AI总结：
本文通过拟等距拟作用建立可数无限近似群的几何理论，证明几何有限生成近似群具有典范拟等距类型，并研究渐近维数、增长类型及双曲空间上的基本性质。
摘要：
我们发展了可数无限近似群的几何理论基础，扩展了 Björklund 和第二作者的工作。我们的理论基于近似群在度量空间上的拟等距拟作用（qiqac）这一概念。更具体地说，我们引入了近似群有限生成的几何概念，并证明每个几何有限生成的近似群都在一个恰当测地度量空间上允许一个几何 qiqac。然后我们证明所有这样的空间都是拟等距的，因此可以用来为每个几何有限生成的近似群关联一个典范的 QI 类型。这反过来又使我们能够利用度量空间的 QI 不变量来定义近似群的几何不变量。我们考虑的不变量包括渐近维数、有限性性质、端数和增长类型。对于具有多项式增长的几何有限生成近似群，我们基于 Hrushovski 以及 Breuillard--Green--Tao 的工作推导出 Gromov 多项式增长定理的一个版本。一个特别的焦点是双曲空间上的 qiqac。我们最强的结果是对那些在恰当测地双曲空间上允许几何 qiqac 的近似群得到的。对于这样的“双曲近似群”，我们类比双曲群的情形建立了一系列基本性质。例如，我们证明它们的渐近维数比其 Gromov 边界的拓扑维数大一，并且——在某种“非初等”的温和假设下——它们具有指数增长并极小地作用在其 Gromov 边界上。我们还研究了双曲空间上的凸余紧 qiqac。利用 Morse 边界理论，我们将关于双曲空间上 qiqac 的部分结果推广到具有非平凡 Morse 边界的恰当测地度量空间上的 qiqac。
英文摘要：
We develop the foundations of a geometric theory of countably-infinite approximate groups, extending work of Björklund and the second-named author. Our theory is based on the notion of a quasi-isometric quasi-action (qiqac) of an approximate group on a metric space. More specifically, we introduce a geometric notion of finite generation for approximate group and prove that every geometrically finitely-generated approximate group admits a geometric qiqac on a proper geodesic metric space. We then show that all such spaces are quasi-isometric, hence can be used to associate a canonical QI type with every geometrically finitely-generated approximate group. This in turn allows us to define geometric invariants of approximate groups using QI invariants of metric spaces. Among the invariants we consider are asymptotic dimension, finiteness properties, numbers of ends and growth type. For geometrically finitely-generated approximate groups of polynomial growth we derive a version of Gromov's polynomial growth theorem, based on work of Hrushovski and Breuillard--Green--Tao. A particular focus is on qiqacs on hyperbolic spaces. Our strongest results are obtained for approximate groups which admit a geometric qiqac on a proper geodesic hyperbolic space. For such "hyperbolic approximate groups" we establish a number of fundamental properties in analogy with the case of hyperbolic groups. For example, we show that their asymptotic dimension is one larger than the topological dimension of their Gromov boundary and that - under some mild assumption of being "non-elementary" - they have exponential growth and act minimally on their Gromov boundary. We also study convex cocompact qiqacs on hyperbolic spaces. Using the theory of Morse boundaries, we extend some of our results concerning qiqacs on hyperbolic spaces to qiqacs on proper geodesic metric spaces with non-trivial Morse boundary.
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