Analytic equivalence relations and bi-embedability

TL;DR

This paper demonstrates that bi-embeddability relations for various classes of countable structures are as complex as any analytic equivalence relation, showing their maximal complexity under Borel reducibility.

Contribution

It proves that every analytic equivalence relation is Borel equivalent to a bi-embeddability relation, extending previous results and answering open questions about the relation's complexity.

Findings

Bi-embeddability relations are complete for analytic equivalence relations.

All analytic equivalence relations are Borel equivalent to bi-embeddability relations.

Results apply to diverse classes including metric spaces and Banach spaces.

Abstract

Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of L_{\omega_1 \omega}) is far from complete (see [5, 2]). In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of L_{\omega_1 \omega}, but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.