Logic for metric structures and the number of universal sofic and hyperlinear groups

TL;DR

This paper employs model theory of metric structures to provide an alternative proof that, under the failure of the Continuum Hypothesis, there are many non-isomorphic universal sofic and hyperlinear groups, answering a question by Thomas.

Contribution

It introduces a new model-theoretic approach to analyze the diversity of universal sofic and hyperlinear groups, extending previous results.

Findings

Existence of many non-isomorphic universal sofic groups under CH failure

Extension of the method to hyperlinear groups

Positive answer to Thomas's question

Abstract

Using the model theory of metric structures, I give an alternative proof of the following result by Thomas: If the Continuum Hypothesis fails then there are power of the continuum many universal sofic groups up to isomorphism. This method is also applicable to universal hyperlinear groups, giving a positive answer to a question posed by Thomas.