Title

Linearly ordered structures and conjugacy classes of their automorphism groups

Abstract

In the talk we address the following (still open) question: Does there exist a Polish non-archimedean group (equivalently: automorphism group of an ultrahomogeneous structure) that is extremely amenable and has ample generics? Every such automorphism group would have to preserve a linear order. In fact, it is unknown if there exists a linearly ordered ultrahomogeneous structure whose automorphism group has a comeager 2-dimensional diagonal conjugacy class.

We provide general conditions which imply that the automorphism group of an ordered ultrahomogeneous structure has no comeager conjugacy class or comeager 2-dimensional diagonal conjugacy class. Moreover, we provide new examples of ordered ultrahomogeneous structures, whose automorphism groups have a comeager conjugacy class. These are the universal ordered boron tree and the universal ordered poset.

This is joint work with Maciej Malicki.

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