$L^2$-Betti numbers and costs in the framework of discrete groupoids

TL;DR

This paper unifies the theories of $L^2$-Betti numbers and costs within the framework of probability measure preserving discrete groupoids, providing a comprehensive theoretical foundation.

Contribution

It introduces a unified framework for $L^2$-Betti numbers and costs in the context of discrete groupoids, bridging existing theories.

Findings

Unified the theories of $L^2$-Betti numbers and costs

Established foundational results for groupoids

Provided a comprehensive theoretical framework

Abstract

We unify the known basic theories on $L^2$-Betti numbers and costs in the framework of probability measure preserving discrete groupoids.