拓扑群、μ-型及其稳定子
Topological groups, μ-types and their stabilizers
摘要：
我们考虑一个任意拓扑群$G$可定义在一个结构$\mathcal M$中，使得$G$的拓扑的某些基由可定义在$\mathcal M$中的集合组成。对于每个这样的群$G$，我们关联一个部分类型的紧致$G$ -空间$S^μ_G(M)=\{p_μ:p\in S_G(M)\}$，它是通常类型空间$S_G(M)$的商，由两个类型的关系“无限接近彼此”。在o-最小设置中，如果$p$是一个可定义类型，那么它有一个对应的可定义子群$Stab_μ(p)$，它是$p_μ$的稳定子。当$p$在$\mathcal M$的意义下是无界的时，这个群是非平凡的;事实上它是一个无挠可解群。一路上，我们分析了一般结构的$S^μ_G(M)$和它的连接到塞缪尔紧化的拓扑群。
英文摘要：
We consider an arbitrary topological group $G$ definable in a structure $\mathcal M$, such that some basis for the topology of $G$ consists of sets definable in $\mathcal M$. To each such group $G$ we associate a compact $G$-space of partial types $S^μ_G(M)=\{p_μ:p\in S_G(M)\}$ which is the quotient of the usual type space $S_G(M)$ by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if $p$ is a definable type then it has a corresponding definable subgroup $Stab_μ(p)$, which is the stabilizer of $p_μ$. This group is nontrivial when $p$ is unbounded in the sense of $\mathcal M$; in fact it is a torsion-free solvable group. Along the way, we analyze the general construction of $S^μ_G(M)$ and its connection to the Samuel compactification of topological groups.
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