On some permanence properties of exact groupoids

TL;DR

This paper investigates the permanence properties of exactness in locally compact groupoids, showing how exactness is preserved under certain subgroupoids and actions, and exploring related concepts like inner exactness.

Contribution

It extends known results on exact groups to groupoids, demonstrating that exactness descends to specific subgroupoids and actions, and provides new insights into inner exactness.

Findings

Exactness is preserved under certain closed subgroupoids.

Transformation groupoids from exact groupoid actions are exact.

Partial converse generalizes Kirchberg and Wassermann's theorem.

Abstract

A locally compact groupoid is said to be exact if its associated reduced crossed product functor is exact. In this paper, we establish some permanence properties of exactness, including generalizations of some known results for exact groups. Our primary goal is to show that exactness descends to certain types of closed subgroupoids, which in turn gives conditions under which the isotropy groups of an exact groupoid are guaranteed to be exact. As an initial step toward these results, we establish the exactness of any transformation groupoid associated to an action of an exact groupoid on a locally compact Hausdorff space. We also obtain a partial converse to this result, which generalizes a theorem of Kirchberg and Wassermann. We end with some comments on the weak form of exactness known as inner exactness.