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We prove that every finitely generated group with recursive asphericalpresentation embeds into a group with finite aspherical presentation. This andseveral known facts about groups and manifolds imply that there exists a4-dimensional closed aspherical manifold $M$ such that the fundamental group$\pi_1(M)$ coarsely contains an expander, and so it has infinite asymptoticdimension, is not coarsely embeddable into a Hilbert space, and does notsatisfy the Baum-Connes conjecture with coefficients. Closed asphericalmanifolds with any of these properties were previously unknown.

Aspherical groups and manifolds with extreme properties