On the quantitative coarse Baum-Connes conjecture with coefficients

TL;DR

This paper introduces a refined version of the coarse Baum-Connes conjecture with coefficients, called QCBC, and demonstrates its derivation from the original conjecture, providing new insights and examples in coarse geometry.

Contribution

The paper defines the QCBC, proves its relation to the coarse Baum-Connes conjecture with coefficients, and reduces QCBC to a uniformly quantitative version for bounded metric spaces.

Findings

QCBC is derived from the coarse Baum-Connes conjecture with coefficients.

Many examples satisfy QCBC.

QCBC can be reduced to a uniformly quantitative conjecture for bounded metric spaces.

Abstract

In this paper, we introduce the quantitative coarse Baum-Connes conjecture with coefficients (or QCBC, for short) for proper metric spaces which refines the coarse Baum-Connes conjecture. And we prove that QCBC is derived by the coarse Baum-Connes conjecture with coefficients which provides many examples satisfying QCBC. In the end, we show QCBC can be reduced to the uniformly quantitative coarse Baum-Connes conjecture with coefficients of a sequence of bounded metric spaces.