论文标题
超越拉斯卡集团
Beyond the Lascar Group
论文作者
论文摘要
我们在一阶设置中工作，其中结构分布在公制空间上，仅量化仅限制子集。假设对度量空间的双倍属性，我们定义了与该理论相关的规范{\ em core} $ \ Mathcal {J} $，该理论是一种局部紧凑的结构，该结构嵌入了任何模型上的类型空间。 $ \ Mathcal {J} $的自动形态组，Modulo某些Infinitessimal Automorthisms，是一个本地紧凑的组$ \ Mathcal {G} $。该理论模型的自态群体与$ \ Mathcal {g} $相关，而不是通过同态的，而是通过{\ em quasi-quasi-homormormormormormormormorlism}相关，尊重乘法到某些规范的紧凑型误差集。这种基本结构用于描述近似亚组的性质。具体来说，我们获得了$ sl_n（{\ Mathbb {r}}）$或$ sl_n（{\ Mathbb {q}} _ p）$的（{\ Mathbb {r}}）$（{\ Mathbb {r}}）的（{\ Mathbb {r}}}）的完整分类。
We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical {\em core} $\mathcal{J}$ associated to such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of $\mathcal{J}$, modulo certain infinitesimal automorphisms, is a locally compact group $\mathcal{G}$. The automorphism groups of models of the theory are related with $\mathcal{G}$, not in general via a homomorphism, but by a {\em quasi-homomorphism}, respecting multiplication up to a certain canonical compact error set. This fundamental structure is applied to describe the nature of approximate subgroups. Specifically we obtain a full classification of (properly) approximate lattices of $SL_n({\mathbb{R}})$ or $SL_n({\mathbb{Q}}_p)$.
