Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups

Turbulence, amalgamation and generic automorphisms of homogeneous structures

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-Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which…

某些高度齐次置换群的不变子集中的通有元素

Generics in invariant subsets of some highly homogeneous permutation groups

本文证明不涉及循环序的闭高度齐次置换群中，共轭类闭包含余稠密共轭类，且有限部分映射族具共尾合并性质，并推广至典型超齐次偏序自同构。

Ordered structures and large conjugacy classes

This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraisse limit) that is extremely amenable, and has ample generics. As Fraisse limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraisse limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager $2$-dimensional diagonal conjugacy class. We also provide general conditions im…

Cantor 极小系统的共轭的 Borel 复杂度

The Borel complexity of conjugacy for Cantor minimal systems

本文证明 Cantor 极小系统共轭的 Borel 复杂度与可数图同构相同，通过编码 pro-有限群和逆向极限构造回答 Foreman 问题。

有理Gurarii空间及其线性等距群

The rational Gurarii space and its linear isometry group

研究显示有限维多面体空间和有理有限维多面体空间的偏等距类不具有弱 amalgamation性质，从而推得有理Gurarii空间的线性等距群无来摩conjucate类。方法还表明有限维多面体空间和有限维有理多面体空间不具有Hrushovski性质。

The structure of random automorphisms of countable structures

In order to understand the structure of the `typical' element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different …