Admissible Higson-Roe sequences for transformation groupoids

TL;DR

This paper develops a universal Higson-Roe six-term exact sequence for transformation groupoids of finitely generated groups acting on compact spaces, extending previous results and exploring implications for the Baum-Connes conjecture.

Contribution

It introduces a universal Higson-Roe sequence for transformation groupoids, generalizing the maximal sequence and linking to the rectified Baum-Connes conjecture.

Findings

Constructed a universal Higson-Roe sequence for transformation groupoids.

Generalized the maximal Higson-Roe sequence to these groupoids.

Derived rigidity consequences under the rectified Baum-Connes conjecture.

Abstract

Given a finitely generated discrete group {\Gamma}, we construct for any admissible crossed product completion and for any metrizable finite dimensional compact {\Gamma}-space X, a universal Higson-Roe six-term exact sequence for the transformation groupoid X\rtimes {\Gamma}. In particular, we generalize the maximal Higson- Roe sequence to such groupoids. In the case where the groupoid X\rtimes {\Gamma} satisfies the rectified Baum-Connes conjecture, this yields some rigidity consequences.