You need to know: Symmetric difference of sets and , notation for the number of elements in finite set , notation for set product, set of natural numbers, metric space where is a set and is a metric, infinite dimensional Hilbert space , group, generating set of a group, finitely generated group, word metric of a group with respect to a generating set.
Background: We say that metric space is coarsely embeddable into metric space is there exists a map such that if and only if for all sequences and in . We say that a finitely generated group is coarsely embeddable into if it is so for the word metric with respect to a finite generating set . This property does not depend on the choice of .
A metric space is called uniformly discrete if there exists a constant such that, for any , we have either or . A uniformly discrete metric space is coarsely amenable if for every and , there exist a constant and a collection of finite subsets , for every , such that (a) when , and (b) . A finitely generated group is called coarsely amenable if it is so for the word metric with respect to a finite generating set , and is called coarsely non-amenable otherwise.
The Theorem: On 19th June 2014, Damian Osajda submitted to arxiv a paper in which he proved the existence of finitely generated coarsely non-amenable groups that are coarsely embeddable into the infinite dimensional Hilbert space .
Short context: The notion of coarse amenability is a week version of amenability, and has many equivalent formulations and applications. In particular, it is known that any finitely generated coarsely amenable groups is coarsely embeddable into . The question whether the converse is true (Are groups coarsely embeddable into coarsely amenable?) is a natural question raised by a number of researchers. The Theorem provides a negative answer.
Links: Free arxiv version of the original paper is here, journal version is here.
Theorems of the 21st century 