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A torsion-free counterexample to reduced Baum–Connes injectivity
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Abstract
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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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- Rufus Willett, Guoliang Yu, Higher index theory for certain expanders and Gromov monster groups II (arXiv:1012.4151, 2010). Studies maximal coarse assembly for large-girth expanders, with applications to Baum–Connes for Gromov monster groups; this may bear on the paper's contrast between maximal assembly, which detects the class h, and reduced assembly, which kills it (only part I is cited).
- Paolo Antonini, Sara Azzali, Georges Skandalis, The Baum–Connes conjecture localised at the unit element of a discrete group (arXiv:1807.05892, Compositio Mathematica). Defines a trace-localised assembly map μ_τ whose conjecture still implies strong Novikov; this may bear on the paper's discussion of which Novikov-type formulations the kernel class does or does not refute.
Digests and expositions
- Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics (arXiv:2610.17090, math.HO, 2026-10-12). Section 9.4.2 summarizes this paper's construction (Heisenberg voltage tower, parity-swindle Bott cancellation, sheet-cone complex) and notes that it "asserts finite generation; it establishes neither finite presentation nor its failure" and that the three papers share one architecture. [family-level]
- Open Mathematics Courses, A torsion-free counterexample to reduced Baum–Connes injectivity (online course, October 2026). This 18-lesson exposition, attributed to Claude Opus 5.5 and marked as not independently reviewed, develops the claimed torus-class detector and reduced Bott cancellation. [AI-generated per source]
News and blogs
- Anoushka Nagle and Haley Rodriguez, Dartmouth mathematicians weigh artificial intelligence’s role in the future of their work (The Dartmouth, 2026-10-20). The article discusses the three counterexample manuscripts through an interview with Erik van Erp, who reports stopping after the introductions because he found the papers difficult to read; this is not a proof assessment. [family-level]
Elsewhere
- Emergent Mind: family 285
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026,
author = {{OpenAI}},
title = {{A torsion-free counterexample to reduced Baum--Connes injectivity}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf}{OAI:A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026}},
year = {2026}
}Family 285: Counterexamples to Baum–Connes and Kadison–Kaplansky
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.
Other papers in this family:
- A torsion-free counterexample to the Kadison–Kaplansky projection conjecture · September 23, 2026
- An irrational-trace counterexample to reduced Baum–Connes · September 23, 2026