Coarse embeddability into Hilbert space for CAT(0)-spaces and Helly groups remains unknown

Determine whether every CAT(0)-space and every Helly group, equipped with their natural metrics (e.g., the word length metric for groups), admits a coarse embedding into Hilbert space.

Background

The paper recalls broad classes known to admit coarse embeddings (hyperbolic spaces, mapping class groups, certain linear groups) and uses such embeddings to establish instances of the conjecture. However, for CAT(0)-spaces and Helly groups—central classes in geometric group theory—the existence of coarse embeddings into Hilbert space is not known.

Resolving this would strengthen the geometric route via the Dirac–dual–Dirac method, potentially extending the range of spaces for which the coarse Baum-Connes conjecture (and its filtered-coefficient variant) is known to hold.