# Ordered structures and large conjugacy classes

## TL;DR

This paper investigates the properties of automorphism groups of ordered Fraisse limits, demonstrating the existence of groups with large conjugacy classes and establishing conditions for their absence, with implications for group dynamics.

## Contribution

It provides new results on the conjugacy class structure of automorphism groups of ordered Fraisse limits, including the universal ordered boron tree and poset.

## Key findings

- Automorphism groups of certain ordered Fraisse limits have a comeager conjugacy class.
- These groups lack a comeager 2-dimensional diagonal conjugacy class.
- General conditions are given for the non-existence of large conjugacy classes in such groups.

## 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.