A generalization of Kechris-Rosendal characterizes comeager diagonal conjugacy classes in arbitrary Polish permutation groups, and ball-preserving groups of ordered ultrametric spaces are shown to have a generic element but no generic pair.
Ordered structures and large conjugacy classes
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abstract
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 implying that there is no comeager conjugacy class, comeager $2$-dimensional diagonal conjugacy class or non-meager $2$-dimensional topological similarity class in the automorphism group of an ordered Fraisse limit. We provide a number of applications of these results.
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math.LO 1years
2019 1verdicts
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Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups
A generalization of Kechris-Rosendal characterizes comeager diagonal conjugacy classes in arbitrary Polish permutation groups, and ball-preserving groups of ordered ultrametric spaces are shown to have a generic element but no generic pair.