Measured groupoids beyond equivalence relations and group actions

TL;DR

This paper introduces the first examples of ergodic discrete measured groupoids that are not equivalent to any known group action or relation, expanding the understanding of groupoid structures.

Contribution

It constructs the first genuine ergodic discrete measured groupoids beyond equivalence relations and transformation groupoids using a novel group construction.

Findings

Existence of ergodic discrete measured groupoids not isomorphic to any equivalence relation.

Use of uncountable non-isomorphic 2-generated groups in the construction.

Demonstration of the diversity of measured groupoid structures.

Abstract

We construct the first examples of genuine ergodic discrete measured groupoids that are not isomorphic to any equivalence relation or transformation groupoid. We use a construction due to B.H. Neumann of an uncountable family of pairwise non-isomorphic 2-generated groups for our result.