Aspherical groups and manifolds with extreme properties
arXiv:1103.3873
Abstract
We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional closed aspherical manifold such that the fundamental group coarsely contains an expander, and so it has infinite asymptotic dimension, is not coarsely embeddable into a Hilbert space, and does not satisfy the Baum-Connes conjecture with coefficients. Closed aspherical manifolds with any of these properties were previously unknown.
23 pages. This is the first version. Comments are welcome. v2: This is a preliminary version of the paper. Comparing to v1 many changes are made, proof is expanded and corrected, references and open problems added, 35 pages. Comments are still welcome