Frucht's theorem in Borel setting

TL;DR

This paper extends Frucht's theorem to Borel and topological settings, showing that any standard Borel group can be represented as a Borel automorphism group of a Borel graph, and similarly for Polish groups in a topological context.

Contribution

It proves that any standard Borel group can be realized as a Borel automorphism group of a Borel graph, and any Polish group as the automorphism group of a $oldsymbol{ riangle^0_2}$-graph on a Polish space.

Findings

Any standard Borel group can be realized as a Borel automorphism group of a Borel graph.

Any Polish group can be realized as the homeomorphic automorphism group of a $oldsymbol{ riangle^0_2}$-graph.

The constructions work in both Borel and topological settings.

Abstract

In this paper, we show that Frucht's theorem holds in Borel setting. More specifically, we prove that any standard Borel group can be realized as the Borel automorphism group of a Borel graph. A slight modification of our construction also yields the following result in topological setting: Any Polish group can be realized as the homeomorphic automorphism group of a $\mathbf{\Delta^0_2}$-graph on a Polish space.