# Counterexamples to Baum–Connes and Kadison–Kaplansky > Claims torsion-free groups that break the Kadison–Kaplansky projection conjecture and reduced Baum–Connes. Plain-English explainer with sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-285/ Last updated: 2026-10-14 **Claims torsion-free groups that break the Kadison–Kaplansky projection conjecture and reduced Baum–Connes.** **Trust:** Claimed, not checked · **Papers:** 3 · **First paper:** September 23, 2026 · **Field:** Operator algebras · **Entry:** 285 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. To each group one can attach an algebra of operators, the reduced group C*-algebra. Two famous conjectures predict what this algebra looks like for well-behaved groups. Three companion papers claim to build groups where these predictions fail. One group has no elements of finite order, yet its algebra holds a projection other than 0 and 1. Another group's algebra holds a projection whose trace is irrational, which the Baum–Connes conjecture forbids. ## At a glance - **First step:** 1982 (Paul Baum and Alain Connes, manuscript formulating the conjecture) - **Significance:** Landmark. Baum–Connes and Kadison–Kaplansky are central conjectures linking group geometry and operator algebras; counterexamples would reshape the field. - **Trust:** Claimed, not checked - **Field:** Operator algebras (3 papers) ## The explanation Start with a group that has two elements, 1 and g, where g times g is 1. Build formal sums a·1 + b·g with complex numbers a and b. Now take e = (1 + g)/2. Squaring gives (1 + 2g + g²)/4 = (2 + 2g)/4, which is e again. An element equal to its own square is a projection. The trace of a sum is its coefficient on 1, so e has trace 1/2. This projection exists because g has finite order. An element of finite order other than 1 is called torsion. Torsion-free groups seem to behave differently. Take the integers under addition. Their operator algebra turns out to be the continuous functions on a circle. A projection is then a function taking only the values 0 and 1. On a connected circle, such a function must be constant. So the only projections are 0 and 1. The Kadison–Kaplansky conjecture says this holds for every torsion-free group. The algebra is the reduced group C*-algebra, the closure of the group's sums acting on square-summable sequences. Pimsner and Voiculescu proved the conjecture for free groups[1]. A main route to it is the Baum–Connes conjecture. That conjecture says an assembly map, from the group's topology to the algebra's K-theory, is an isomorphism[2]. K-theory sorts projections into classes. Baum and Connes circulated it in 1982 and published it in 2000[3]. It is known for large classes of groups, including all countable amenable groups[4]. Earlier counterexamples of Higson, Lafforgue and Skandalis used extra coefficients. They do not break the coefficient-free form[5]. The obstacle was to find a group, with no coefficients, whose algebra escapes the topology. The three papers claim to do that. The first builds a finitely generated torsion-free group whose assembly map sends a class of infinite order to zero[6]. The group is finitely generated, but finite presentation is not part of the conclusion[7]. It disproves neither maximal rational injectivity nor the classical Novikov conjecture[8]. The second builds a finitely generated torsion-free group with a projection other than 0 and the identity[9]. Its trace lies strictly between 0 and 1/2. That projection is in the completed algebra, not in the group ring of finite sums[10]. The third uses Lück's theorem: every assembly class has a rational trace[11]. It then builds a finitely generated group with a projection of irrational trace[12]. That projection cannot come from assembly. The constructions use expander graphs, sparse networks that are still very well connected. How sure is this? A computer proof checker called Lean has not checked any part of it. Each claim rests on a long chain of graph, spectral and operator estimates. No specialist review of the papers has been published. **What it doesn't show.** This does not disprove the classical Novikov conjecture or maximal rational injectivity, and it gives no finitely presented example. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Landmark if it holds: these are counterexamples to two central conjectures. There is no Lean check and no published specialist review, so the claims are unconfirmed. ## Words used - **Torsion-free group:** A group in which no element other than the identity returns to the identity after finitely many repeats. - **Projection:** An element of an algebra equal to its own square and to its own adjoint, like e = (1 + g)/2 above. - **Trace:** Here, the coefficient on the identity element; for the example e it is 1/2. - **Reduced group C*-algebra:** The algebra of operators made by letting the group act on square-summable sequences and closing up in norm. - **K-theory:** A way of sorting projections in an algebra into classes that can be added. - **Assembly map:** A map from topological data about the group to the K-theory of its reduced C*-algebra. - **Finitely generated:** Every element is a product of finitely many fixed generators and their inverses. - **Expander graph:** A sparse network of points and links that is still very well connected. ## The original papers (3) - [A torsion-free counterexample to reduced Baum–Connes injectivity](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf) - September 23, 2026 - [A torsion-free counterexample to the Kadison–Kaplansky projection conjecture](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf) - September 23, 2026 - [An irrational-trace counterexample to reduced Baum–Connes](https://github.com/openai/math/blob/main/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf) - September 23, 2026 ## Quoted lines from the paper (12) [1] From [A torsion-free counterexample to the Kadison–Kaplansky projection conjecture](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf): > Pimsner and Voiculescu proved the free-group case by computing the \(K\)-theory of reduced crossed products by free groups. [2] From [A torsion-free counterexample to reduced Baum–Connes injectivity](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf): > The Baum--Connes conjecture asserts that this map is an isomorphism. [3] From [A torsion-free counterexample to reduced Baum–Connes injectivity](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf): > Baum and Connes introduced this index-theoretic comparison in a preprint circulated in 1982 and published in 2000. [4] From [A torsion-free counterexample to reduced Baum–Connes injectivity](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf): > Higson and Kasparov prove it with separable coefficients for groups acting properly by affine isometries on Hilbert space, including countable amenable groups. [5] From [An irrational-trace counterexample to reduced Baum–Connes](https://github.com/openai/math/blob/main/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf): > Those constructions do not give a counterexample to the coefficient-free group assertion above. [6] From [A torsion-free counterexample to reduced Baum–Connes injectivity](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf): > Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order. [7] From [A torsion-free counterexample to reduced Baum–Connes injectivity](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf): > The group is finitely generated; finite presentation is not part of the conclusion. [8] From [A torsion-free counterexample to reduced Baum–Connes injectivity](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf): > Our result disproves neither maximal rational injectivity nor this classical assertion; in particular, higher signatures associated to low-degree cohomology already have the homotopy invariance established in. [9] From [A torsion-free counterexample to the Kadison–Kaplansky projection conjecture](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf): > Theorem … resolves the projection form of the Kadison--Kaplansky conjecture negatively. [10] From [A torsion-free counterexample to the Kadison–Kaplansky projection conjecture](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf): > Thus the projection \(e\) lies in the reduced completion but not in the group ring. [11] From [An irrational-trace counterexample to reduced Baum–Connes](https://github.com/openai/math/blob/main/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf): > L\"uck's trace theorem implies that its value on every degree-zero assembly class is rational. [12] From [An irrational-trace counterexample to reduced Baum–Connes](https://github.com/openai/math/blob/main/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf): > There exist a finitely generated discrete group \(\Gsur\) and a projection \(b_0\in C_r^*(\Gsur)\) with irrational canonical trace. ## Other sources - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [Rigidity and arithmetic of lattice von Neumann algebras](https://closedform.nishachay.tech/e/openai-math-2026-286/) - Operator algebras - [Relative bicentralizers and modular spectral recovery](https://closedform.nishachay.tech/e/openai-math-2026-290/) - Operator algebras - [Isomorphism of the free group factors](https://closedform.nishachay.tech/e/openai-math-2026-287/) - Operator algebras - [Kadison's similarity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-288/) - Operator algebras - [Strong Kadison–Kastler stability and its spatial boundaries](https://closedform.nishachay.tech/e/openai-math-2026-289/) - Operator algebras - [Operator algebras](https://closedform.nishachay.tech/science/mathematics/operator-algebras/) - Field · 19 results - [Open since the 1980s](https://closedform.nishachay.tech/open-since/1980s/) - Era · 60 results - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results - [Landmark results](https://closedform.nishachay.tech/significance/landmark/) - Significance · 49 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20285%5D%20)