Cost of inner amenable groupoids

TL;DR

This paper extends the concept of inner amenability to discrete p.m.p. groupoids and proves that such groupoids have fixed cost 1 for all principal extensions, unifying key results in the theory of cost and amenability.

Contribution

It introduces the notion of inner amenability for groupoids and proves they have fixed cost 1, generalizing previous results for groups and equivalence relations.

Findings

Inner amenable groupoids have fixed cost 1.

Generalizes Kechris's theorem on cost and ergodic relations.

Unifies results on fixed price for groups and relations.

Abstract

Kida and Tucker-Drob recently extended the notion of inner amenability from countable groups to discrete p.m.p. groupoids. In this article, we show that inner amenable groupoids have "fixed priced 1" in the sense that every principal extension of an inner amenable groupoid has cost 1. This simultaneously generalizes and unifies two well known results on cost from the literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p. equivalence relation admitting a nontrivial asymptotically central sequence in its full group has cost 1, and (2) a theorem of Tucker-Drob stating that inner amenable groups have fixed price 1.