Controlled K-theory for groupoids and applications to Coarse Geometry

TL;DR

This paper introduces controlled K-theory, a powerful tool for analyzing the K-theory of groupoid crossed products, and applies it to establish new results in coarse geometry related to the Baum-Connes conjecture.

Contribution

It generalizes quantitative K-theory to controlled K-theory, enabling the study of crossed products by étale groupoids and quantum groups, and relates the controlled Baum-Connes conjecture to classical and coarse versions.

Findings

Proves the maximal controlled coarse Baum-Connes conjecture for spaces with fibred coarse embeddings.

Defines controlled assembly maps that factorize the Baum-Connes assembly maps.

Relates the controlled conjecture for groupoids to classical and coarse Baum-Connes conjectures.

Abstract

We develop a generalization of quantitative $K$-theory, which we call controlled $K$-theory. It is powerful enough to study the $K$-theory of crossed product of $C^*$-algebras by action of \'etale groupoids and discrete quantum groups. In this article, we will use it to study groupoids crossed products. We define controlled assembly maps, which factorize the Baum-Connes assembly maps, and define the controlled Baum-Connes conjecture. We relate the controlled conjecture for groupoids to the classical conjecture, and to the coarse Baum-Connes conjecture. This allows to give applications to Coarse Geometry. In particular, we can prove that the maximal version of the controlled coarse Baum-Connes conjecture is satisfied for a coarse space which admits a fibred coarse embedding, which is a stronger version of a result of M. Finn-Sell.