Actions of Amenable Equivalence Relations on CAT(0) Fields

TL;DR

This paper investigates how amenable equivalence relations act on Borel fields of CAT(0) spaces, establishing a rigidity result that extends known group action results to a broader equivalence relation context.

Contribution

It introduces the concept of equivalence relation actions on Borel fields of metric spaces and proves a new rigidity theorem for such actions on CAT(0) spaces.

Findings

Established a rigidity theorem for amenable equivalence relation actions on CAT(0) fields.

Extended group action results to the setting of equivalence relations.

Provided foundational definitions for actions on Borel fields of metric spaces.

Abstract

After a short introduction to the general notion of Borel fields of metric spaces we introduce the notion of the action of an equivalence relation on such fields. Then, we specify the study to the Borel fields of proper CAT(0) spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.