Inner amenable groupoids and central sequences

TL;DR

This paper introduces the concept of inner amenability for discrete p.m.p. groupoids, explores its properties and examples, and links it to central sequences in associated algebraic structures, revealing new connections in ergodic theory.

Contribution

It defines inner amenability for groupoids and establishes its relationship with central sequences and orbit equivalence relations, extending previous concepts from groups to groupoids.

Findings

Inner amenability implies certain orbit equivalence relations are inner amenable.

Compact actions of inner amenable groups produce inner amenable orbit relations.

Extensions with stability or non-trivial central sequences are also shown to be inner amenable.

Abstract

We introduce inner amenability for discrete p.m.p. groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations which either are stable or have a non-trivial central sequence in their full group.