The Baum-Connes Conjecture (coefficient-free, reduced)

Three counterexamples. (1) A finitely generated group with a projection in $C_r^*(G)$ of irrational canonical trace, so $\mu_0^G$ is not surjective. (2) A finitely generated torsion-free group whose degree-zero reduced assembly map kills a class of infinite order, so rational injectivity fails (the reduced form of the strong Novikov conjecture). (3) The Kadison-Kaplansky counterexample also gives a torsion-free group with a class outside the assembly image. Not shown: failure of maximal assembly, a finitely presented counterexample, or failure of the classical Novikov conjecture.

Three counterexamples. (1) A finitely generated group with a projection in $C_r^*(G)$ of irrational canonical trace, so $\mu_0^G$ is not surjective. (2) A finitely generated torsion-free group whose degree-zero reduced assembly map kills a class of infinite order, so rational injectivity fails (the reduced form of the strong Novikov conjecture). (3) The Kadison-Kaplansky counterexample also gives a torsion-free group with a class outside the assembly image. Not shown: failure of maximal assembly, a finitely presented counterexample, or failure of the classical Novikov conjecture.