The coarse Baum-Connes conjecture for Busemann non-positively curved spaces

TL;DR

This paper proves the coarse Baum-Connes conjecture for Busemann non-positively curved spaces without assuming bounded coarse geometry, and demonstrates how to compute related K-theoretic invariants via visual boundaries.

Contribution

It establishes the isomorphism of coarse assembly maps for a broad class of non-positively curved spaces and provides methods to compute coarse K-homology and Roe algebra K-theory.

Findings

Coarse assembly maps are isomorphisms for Busemann non-positively curved spaces.

K-homology and Roe algebra K-theory can be computed using visual boundaries.

The results do not require the spaces to have bounded coarse geometry.

Abstract

We prove that the coarse assembly maps for proper metric spaces which are non-positively curved in the sense of Busemann are isomorphisms, where we do not assume that the spaces are with bounded coarse geometry. Also it is shown that we can calculate the coarse K-homology and the K-theory of the Roe algebra by using the visual boundaries.