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.
Kwiatkowska, Universal minimal flows of generalized Waewski dendrites , J
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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.