Borel graphable equivalence relations

TL;DR

This paper investigates which analytic equivalence relations can be realized as connectedness relations of Borel graphs, exploring their properties, examples, and connections to Polish groups and computability.

Contribution

It characterizes Borel graphable relations, constructs examples and non-examples, and introduces the concept of graphic groups, expanding understanding of Borel graphability in descriptive set theory.

Findings

An equivalence relation from countable admissible ordinals is Borel graphable iff a non-constructible real exists.

All isomorphism relations for countable structures are Borel graphable.

The class of graphic groups includes $S_ Infty$ and all connected Polish groups.

Abstract

This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence relations are Borel graphable. First, we study an equivalence relation arising from the theory of countable admissible ordinals and show that it is Borel graphable if and only if there is a non-constructible real. As a corollary of the proof, we construct an analytic equivalence relation which is (provably in ZFC) not Borel graphable and an effectively analytic equivalence relation which is Borel graphable but not effectively Borel graphable. Next, we study analytic equivalence relations given by the isomorphism relation for some class of countable structures. We show that all such equivalence relations are Borel graphable, which implies that for every Borel action of $S_\infty$, the associated orbit equivalence relation is Borel graphable. This leads us to study the class of Polish groups whose Borel actions always give rise to Borel graphable orbit equivalence relations; we refer to such groups as graphic groups. We show that besides $S_\infty$, the class of graphic groups includes all connected Polish groups and is closed under countable products. We finish by studying structural properties of the class of Borel graphable analytic equivalence relations and by considering two variations on Borel graphability: a generalization with hypegraphs instead of graphs and an analogue of Borel graphability in the setting of computably enumerable equivalence relations.