Extremely amenable automorphism groups of countable structures

TL;DR

This paper demonstrates that there are continuum many non-isomorphic extremely amenable groups as automorphism groups of countable structures, and analyzes their classification complexity within descriptive set theory.

Contribution

It constructs continuum many pairwise non-isomorphic extremely amenable groups and studies their classification complexity in the Borel hierarchy.

Findings

There are continuum many such groups.

The class of extremely amenable groups is Borel.

Their isomorphism relation exceeds complexity of countable structure isomorphisms.

Abstract

In this paper we address the question: How many pairwise non-isomorphic extremely amenable groups are there which are separable metrizable or even Polish? We show that there are continuum many such groups. In fact we construct continuum many pairwise non-isomorphic extremely amenable groups as automorphism groups of countable structures. We also consider this classification problem from the point of view of descriptive set theory by showing that the class of all extremely amenable closed subgroups of $S_\infty$ is Borel and their isomorphism relation is more complex than any isomorphism relation of countable structures in the Borel reducibility hierarchy.