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

TL;DR

This paper explores weak amalgamation in non-archimedean groups, linking conjugacy class properties to structural embedding properties, and characterizes limits of weak Fraïssé classes, with applications to ultrametric spaces.

Contribution

It generalizes previous results on conjugacy classes, characterizes non-homogenizable limits of weak Fraïssé classes, and studies conjugacy classes in ultrametric space groups.

Findings

Polish group $G$ has a comeager $n$-diagonal conjugacy class iff certain embedding properties hold.

Characterization of limits of weak Fraïssé classes that are not homogenizable.

Analysis of conjugacy classes in groups of ball-preserving bijections of ultrametric spaces.

Abstract

We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure $M$, Polish group $G$ of permutations of $M$, and $n \geq 1$, $G$ has a comeager $n$-diagonal conjugacy class iff the family of all $n$-tuples of $G$-extendable bijections between finitely generated substructures of $M$, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fra\"{i}ss\'{e} classes that are not homogenizable. Finally, we investigate $1$- and $2$-diagonal conjugacy classes in groups of ball-preserving bijections of certain ordered ultrametric spaces.