Measured groupoids beyond equivalence relations and group actions

TL;DR

The paper answers whether genuine discrete measured groupoids—those not isomorphic to equivalence relations or transformation groupoids—exist and how to construct them. It leverages B. H. Neumann's uncountable family of pairwise non-isomorphic 2-generated groups, organized into a Borel field, and couples this with a given ergodic equivalence relation via a semidirect product construction to produce genuine ergodic groupoids. The main result shows that for any ergodic equivalence relation on a diffuse standard probability space, there exists a genuine discrete measured groupoid realizing that relation; the construction can be arranged so the groupoid is ICC, yielding a factor von Neumann algebra $L(\mathcal{G})$, and it extends to non-ergodic and type-specific settings. The work also excludes the atomic-unit-case as always corresponding to transformation groupoids and provides a broad framework to realize prescribed type and flow of weights in the associated von Neumann algebras, advancing the understanding of ergodic groupoids beyond standard orbit equivalence and transformation-group constructions.

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.

Theorems & Definitions (19)

• Theorem B
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.5
• Definition 2.6
• Proposition 2.7
• Lemma 3.1
• proof
• Lemma 3.3