1 · The essentials
These manuscripts challenge two predictions about how information about a mathematical group is reflected in an associated algebra. One prediction concerns a comparison rule that is expected both to preserve information and to reach every intended output. The main manuscript claims an example where this rule loses information that remains significant even when rational fractions are allowed. A companion manuscript claims a separate example where the rule misses an output. Another companion challenges a prediction that certain groups should have no intermediate switch-like elements in their associated algebra: only completely off or completely on. The first two constructions use groups with a finite set of generators and no nonidentity element that returns to the identity after finitely many repetitions. These are claims reported in the supplied abstracts; the main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, think of a system that translates a catalogue into a warehouse inventory. One claimed failure is that a meaningful catalogue entry disappears during translation; another is that a warehouse item cannot be obtained from any catalogue entry. This illustrates the two failures of the comparison rule, not the details of the mathematical constructions.
Why it matters
The claimed examples challenge both information preservation and completeness for the coefficient-free reduced Baum–Connes assembly map. The projection example separately challenges the Kadison–Kaplansky prediction even for a finitely generated torsion-free group. These are distinct conclusions, and the supplied abstracts do not identify their example groups with one another.
2 · Going further
The central object is the coefficient-free reduced Baum–Connes assembly map. Manuscript 1 claims a finitely generated torsion-free discrete group for which the degree-zero assembly map has a kernel class of infinite order. A kernel class is sent to zero; infinite order means that no nonzero integer multiple of that class vanishes. This is the stated witness to failure of rational injectivity, rather than merely an obstruction that disappears upon allowing rational coefficients.
Manuscript 2 addresses the Kadison–Kaplansky projection conjecture. It claims a finitely generated torsion-free discrete group whose reduced group -algebra contains a projection other than and . The abstract presents this construction as a counterexample to that conjecture. It does not state that this is the same group used in Manuscript 1.
Manuscript 3 gives a separate claimed failure of surjectivity. It constructs a finitely generated discrete group whose reduced group -algebra contains a projection with irrational canonical trace, and states that the projection's -class lies outside the image of the coefficient-free reduced assembly map. Unlike the first two abstracts, this abstract does not assert that its group is torsion-free.
The supplied abstracts describe the witnesses to failure—an infinite-order kernel class, a nontrivial projection, and an irrational-trace projection class outside the image—but provide no construction details or intermediate argument. Thus they explain what each proposed counterexample accomplishes, not how the groups or projections are built. Together, they report two distinct failures of assembly and a separate failure of the projection conjecture.
Useful notions
- Groups and generators
- Torsion-free groups
- Kernels and images of maps
- Reduced group C*-algebras
- Projections and K-theory
What was known before / what is new
Before
The source identifies the coefficient-free reduced Baum–Connes conjecture and the Kadison–Kaplansky projection conjecture as the predictions being challenged. It supplies no history, earlier partial results, or prior construction methods.
New
Manuscript 1 claims failure of rational injectivity witnessed by an infinite-order degree-zero kernel class for a finitely generated torsion-free discrete group. Manuscript 2 claims a nontrivial projection for a group with the same stated properties. Manuscript 3 claims an irrational-trace projection whose -class is outside the assembly image, giving a separate failure of surjectivity.
Limits and assumptions
The assembly claims concern the coefficient-free reduced version, not other variants. The first and second manuscripts explicitly require finitely generated torsion-free discrete groups; the third specifies a finitely generated discrete group without asserting torsion-freeness. The supplied abstracts do not establish any identification among the example groups and do not provide enough argument to assess the constructions. The main result is not formalized in Lean and has not been formally verified. No further open questions are specified in the source.
Mini glossary
- Torsion-free group —
- A group in which no nonidentity element becomes the identity after a positive finite number of repetitions.
- Finitely generated group —
- A group whose elements can all be formed from a finite collection of generators and their inverses.
- Rational injectivity —
- The property that a map loses no information after rational coefficients are allowed.
- Surjectivity —
- The property that every element of a map's target is the image of some input.
- Projection —
- A self-adjoint element of a C*-algebra satisfying .
- Kernel class —
- A class that a map sends to zero.
Keywords : Baum–Connes assembly · Kadison–Kaplansky conjecture · Group C*-algebras
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
A torsion-free counterexample to reduced Baum–Connes injectivity
23/09/2026We 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.
A torsion-free counterexample to the Kadison–Kaplansky projection conjecture
23/09/2026We 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.
An irrational-trace counterexample to reduced Baum–Connes
23/09/2026We construct a finitely generated discrete group whose reduced group C∗-algebra contains a projection of irrational canonical trace. Its K0-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.