OpenAI research collection · 3 manuscripts · Source checked 2026-10-19
Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C^(∗)-algebra, disproving the Kadison–Kaplansky conjecture.
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We disprove rational injectivity of the coefficient-free reduced Baum–Connes assembly map for torsion-free groups. Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order.
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We construct a finitely generated torsion-free discrete group whose reduced group C^(∗)-algebra contains a projection other than zero and the identity. This disproves the Kadison–Kaplansky projection conjecture.
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We construct a finitely generated discrete group whose reduced group C^(∗)-algebra contains a projection of irrational canonical trace. Its K_(0)-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map. This disproves the coefficient-free reduced Baum–Connes conjecture for countable discrete groups.
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