In the talk, we will address the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a countable structure or of a Fraisse limit) that is extremely amenable and has ample generics? In fact, it is unknown if there exists a linearly ordered structure whose automorphism group has a comeager 2-dimensional diagonal conjugacy class.

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. Moreover, we provide general conditions implying that there is no comeager conjugacy class or comeager 2-dimensional diagonal conjugacy class in the automorphism group of an ordered Fraisse limit.

This is joint work with Maciej Malicki.