论文标题
量化离散组的度量近似值
Quantifying metric approximations of discrete groups
论文作者
论文摘要
我们介绍并系统地研究了一个配置函数，其渐近行为量化了一个有限生成的组$ g $的度量或大小，由$ \ mathcal {f} = \ {（g_α，d_α，d_α，d_α，k__α，k_α，k_α，\ var p var p var p varepsilon_} $ i in i in i in i in i in y in）配备了双重不变的度量$d_α$和尺寸$k_α$，对于严格的正实数$ \ varepsilon_ {α} $，因此$ \ inf_ {α} \ varepsilon_ {α}> 0 $。通过我们介绍的残留符合性概况的概念，我们的方法概括了可正约的组的经典等速度（又名Folner）概况，并最近引入了量化残留有限基团的功能。我们的观点更加笼统，涵盖了多型和沙发近似以及许多其他度量近似值，例如弱的Sofic，弱型成熟和线性的SOFIC近似值。
We introduce and systematically study a profile function whose asymptotic behavior quantifies the dimension or the size of a metric approximation of a finitely generated group $G$ by a family of groups $\mathcal{F}=\{(G_α, d_α, k_α, \varepsilon_{α})\}_{α\in I, }$ where each group $G_α$ is equipped with a bi-invariant metric $d_α$ and a dimension $k_α$, for strictly positive real numbers $\varepsilon_{α}$ such that $\inf_{α}\varepsilon_{α}>0$. Through the notion of a residually amenable profile that we introduce, our approach generalizes classical isoperimetric (aka Folner) profiles of amenable groups and recently introduced functions quantifying residually finite groups. Our viewpoint is much more general and covers hyperlinear and sofic approximations as well as many other metric approximations such as weakly sofic, weakly hyperlinear, and linear sofic approximations.
