Turbulence, amalgamation, and generic automorphisms of homogeneous structures
Turbulence, amalgamation, and generic automorphisms of homogeneous structures
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均质结构的湍流、合并和一般自同构
DOI：
10.1112/plms/pdl007
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发表时间：
2004
影响因子：
1.8
通讯作者：
Christian Rosendal
中科院分区：
文献类型：
--
作者：
A. Kechris;Christian Rosendal
研究了波兰群中共轭类的拓扑性质，重点研究了齐次可数结构的自同构群。我们首先考虑稠密共轭类的存在性（拓扑Rokhlin性质）。然后，我们刻画当一个自同构群承认一个comeager共轭类（回答问题的桁架），并应用这表明，同胚群的康托空间有一个comeager共轭类（回答问题的阿金，赫尔利和肯尼迪）。最后，我们研究波兰组承认comeager共轭类在任何方面（在这种情况下，该集团被称为承认充足的泛型）。我们表明，波兰组有足够的泛型的小指数属性（推广的结果Hodges，Hodkinson，Lascar和Shelah）和任意同态从这样的群体到可分群是自动连续的。此外，在寡纯置换群的情况下，它们具有不可数共尾性和Bergman性质。这些结果特别适用于许多ω-稳定的，ω 0-范畴结构的自同构群和随机图。在这方面，我们还证明了无限对称群S∞有唯一的非平凡可分群拓扑。对于几个有趣的群，我们还建立了Serre的性质（FH）和（FA）。
We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges, Hodkinson, Lascar and Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many ω‐stable, ℵ0‐categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group S∞ has a unique non‐trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA).