论文标题
几何近似群体理论的基础
Foundations of geometric approximate group theory
论文作者
论文摘要
我们开发了一个近似近似群体的几何理论的基础，扩展了Björklund和第二名作者的工作。我们的理论基于公制空间上近似组的准静电准表（QIQAC）的概念。 更具体地说，我们引入了近似群体的有限产生的几何概念，并证明每个几何有限生成的近似群体都在适当的地理测量空间上允许几何QIQAC。然后，我们证明所有这些空间都是准等级的，因此可以用来将规范的Qi类型与每个几何有限生成的近似组相关联。反过来，这使我们能够使用度量空间的Qi不变性定义近似组的几何不变。我们认为不变的是渐近维度，有限特性，末端数量和生长类型。对于多项式生长的几何有限生成的近似基团，我们根据Hrushovski和Breuillard的工作得出了Gromov的多项式生长定理的一种版本 - Green-Green--Tao。 特定的重点是双曲线空间上的QIQAC。对于在适当的测量双曲线空间上接受几何QIQAC的近似组，我们获得了我们最强的结果。对于这种“双曲线近似组”，我们与双曲基团相比建立了许多基本特性。例如，我们表明它们的渐近维度比其Gromov边界的拓扑维度大，并且在某种温和的假设为“非质量”的假设下，它们具有指数级的生长，并且在其Gromov边界上的作用很小。我们还研究双曲线空间上的CoCoCompact Qiqacs。使用摩尔斯界的理论，我们将有关QIQAC在双曲线空间上的QIQAC的一些结果扩展到具有非平凡摩尔斯山边界的适当的地理指标空间上的QIQAC。
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.
