When geometry and operator indices part company: three group-algebra counterexamples

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Family 285 · Operator algebras
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

A group can be studied geometrically, through spaces on which it acts, or analytically, through operators that translate functions on the group. The Baum–Connes conjecture proposes a precise agreement between invariants obtained by these two routes. Geometric index classes should account for every analytic class, and distinct geometric classes should remain distinguishable analytically.

Family 285 contains three claimed counterexamples. One loses a nonzero geometric class. Another constructs an unexpected projection for a group without finite-order elements. The third constructs a projection whose irrational trace prevents it from coming from geometry. All concern the reduced, coefficient-free formulation. These adjectives distinguish the claims from earlier counterexamples to related formulations and from several conjectures that the manuscripts do not disprove.

2. Background and essential definitions

A discrete group $G$ acts on the Hilbert space $\ell^2(G)$ of square-summable functions by translation:

$$(\lambda(g)\xi)(h)=\xi(g^{-1}h).$$

Finite complex linear combinations of these operators form the represented group ring $\mathbb C[G]$. Closing it in operator norm gives the reduced group C-algebra* $C_r^*(G)$. A C*-algebra is an algebra of bounded operators closed under adjoints and norm limits. Norm completion allows operators that no finite group-ring expression describes.

A projection satisfies $p=p^*=p^2$; geometrically it projects onto a closed subspace. The canonical trace is

$$\tau_G(a)=\langle a\delta_e,\delta_e\rangle,$$

where $e$ is the identity and $\delta_e$ its unit point mass. Thus $\tau_G(1)=1$. On a matrix algebra the trace is the unnormalized sum of diagonal traces, not their average.

The group $K_0(A)$ records stable differences of projections over a C*-algebra $A$; stabilization permits matrices of arbitrary finite size. Accordingly, a matrix projection and a projection in $A$ itself are different objects. The latter is sometimes called a scalar projection, meaning a one-by-one matrix over the algebra, not necessarily a scalar multiple of the identity.

For a torsion-free group, whose only finite-order element is its identity, assembly takes the form

$$\mu_r:K_*(BG)\longrightarrow K_*(C_r^*(G)).$$

Here $BG$ is a classifying space for the group and the left side is topological K-homology, built from geometric index data with compact support. For groups with torsion one instead uses equivariant K-homology of the classifying space for proper actions, where stabilizers are finite. The conjecture asserts that assembly is an isomorphism.

“Coefficient-free” means coefficients $\mathbb C$ with trivial action. Allowing a separate algebra carrying a group action gives a stronger conjecture. “Reduced” specifies the regular representation above; the maximal group C*-algebra uses all unitary representations and need not give the same K-theory.

3. Development history and previous partial results

Baum and Connes developed the index-theoretic comparison beginning with a manuscript circulated in 1982. The formulation with Higson made proper actions and general coefficient algebras central to its scope. Positive results cover large classes: Higson–Kasparov’s theorem applies to groups admitting proper affine isometric actions on Hilbert space, while work of Mineyev–Yu and Lafforgue establishes major hyperbolic-group cases.

Higson, Lafforgue and Skandalis constructed influential counterexamples using expander geometry. Their discrete-group counterexamples concern nontrivial coefficients; their groupoid examples concern a different kind of symmetry object. Neither is already the coefficient-free group counterexample claimed here. Exact crossed-product reformulations were subsequently developed to respond to those failures, and are again separate assertions.

The projection question has a parallel history. Kadison and Kaplansky’s conjecture predicts that a torsion-free group’s reduced algebra has no projections other than zero and one. Pimsner–Voiculescu established the free-group case. More generally, for torsion-free groups the canonical trace is integral on the assembly image, so surjectivity of assembly forces scalar projectionlessness.

Other constructions, including Austin’s, produced irrational dimensions of kernels of group-ring operators. These live naturally in a larger group von Neumann algebra, closed in a weaker operator topology. Their kernel projections need not belong to the reduced C*-algebra. That membership issue is decisive: some groups with irrational von Neumann kernel dimensions already satisfy Baum–Connes.

The new manuscripts therefore seek more than unusual kernel dimensions or another coefficient obstruction. They require precise failures inside the coefficient-free reduced-algebra setting.

4. What the new manuscripts claim

M1: a geometric class that disappears

M1 constructs a finitely generated torsion-free group $G_{\mathrm{inj}}$, an embedding $i:\mathbb Z^2\to G_{\mathrm{inj}}$, and an integral degree-two cohomology class detecting the embedded torus. The torus’s Dirac K-homology class gives an infinite-order element $h$ with

$$h\otimes1\ne0,\qquad \mu_r(h)=0.$$

The tensor product with $\mathbb Q$ in the first expression means that the class survives after ignoring torsion. Thus M1 claims failure even of rational injectivity. Analytically, the rank-zero Bott class associated with the torus becomes zero; geometrically, the integral cohomology pairing keeps the original class nonzero. Finite presentation is not claimed.

M2: a forbidden projection without torsion

M2 constructs a finitely generated torsion-free group $G_{\mathrm{proj}}$ and a projection in its reduced algebra satisfying

$$0<\tau_{G_{\mathrm{proj}}}(e)<\frac12.$$

It is therefore neither zero nor the identity, contradicting the projection form of Kadison–Kaplansky. The nonintegral trace also obstructs coefficient-free reduced assembly surjectivity. This is an actual projection in the algebra itself, after a separate step converting a small-trace matrix projection into a scalar one.

M2 cites a companion result that the same group’s complex group ring still has no nontrivial scalar idempotents. The claimed failure occurs in the norm completion; it is not a counterexample to that algebraic group-ring assertion.

M3: an irrational analytic class

M3 constructs a finitely generated discrete group $G_{\mathrm{sur}}$ and a scalar projection $b_0\in C_r^*(G_{\mathrm{sur}})$ with irrational canonical trace. Lück’s trace theorem makes every degree-zero assembly image rational, so

$$[b_0]\notin\operatorname{im}\mu_r.$$

This group is built using a module over a finite field and is not asserted torsion-free. The three manuscripts construct separate groups; the family does not identify one example simultaneously having every listed property.

5. Why the advance matters

Importance: 5/5. If correct, these constructions overturn central predictions linking group geometry, index theory and operator algebras. They claim both kinds of assembly failure: geometric information can disappear, and analytic information can exist without a geometric preimage.

The projection example is particularly concrete. It says that a torsion-free group can acquire a nontrivial orthogonal decomposition inside its completed translation algebra despite the absence of finite-order group elements. The irrational-trace example supplies a numerical obstruction that can be described without calculating the entire K-theory group.

The distinctions also limit the conclusion. A failure of the reduced comparison does not erase the successful theorems for amenable, hyperbolic or other covered groups. Nor does it automatically contradict the maximal assembly formulation or the classical Novikov conjecture about higher signatures of manifolds.

6. The main idea

Create controlled geometry at many scales

All three constructions use graph geometry to manufacture analytic phenomena while controlling the ambient group. Expanders are finite graphs with strong connectivity, reflected in a uniform spectral gap separating constant functions from oscillating ones. Carefully chosen labels let large graph pieces embed in a group’s Cayley graph, the graph describing multiplication by generators.

A major construction issue is simultaneous overlap: controlling one pair of graph copies is insufficient when infinitely many translated copies interact. The manuscripts organize copies so that overlaps and unused edges can be estimated together. The detailed constructions differ between papers; expander terminology alone is not a substitute for those estimates.

Cancel a Bott class while preserving its detector

M1 places localized Bott representatives at successive heights. One homotopy identifies even-height and odd-height families; a second identifies odd heights with positive even heights. Together they identify a family with the same family minus its initial term, forcing that term’s K-class to vanish.

This resembles cancellation in an infinite sequence, but the representatives must be bounded operators and the homotopies must genuinely belong to the relevant norm-algebra construction. A separate geometric ingredient preserves an embedded torus and extends its integral cohomology detector. The analytic cancellation and topological survival are deliberately built together, then retained in a finitely generated subgroup.

Use a spectral gap to obtain an actual projection

M2 constructs a finite matrix group-ring operator with a separated spectral band and very small positive projection trace. A gap makes the band indicator continuous on the spectrum, so continuous functional calculus places its projection in the reduced C*-algebra. Adjoining finitely many free generators then permits compression to a scalar projection of the same trace, while preserving torsion-freeness and finite generation.

M3 uses local finite-field rules to select isolated expander components. Its operator has a kernel consisting of constants on selected components, with a uniform gap on the entire remainder. The trace becomes a series of powers $p^{-k_i}$, where $p$ is a prime and the chosen exponents make the base-$p$ expansion nonperiodic. That yields irrationality. The global gap, rather than the trace calculation alone, puts the kernel projection in the reduced algebra.

7. Scope, limitations, and open questions

These are existence constructions with finite generation, not claims of practical small presentations or useful numerical parameter bounds. M1 explicitly does not assert finite presentation. The group constructions and spectral realizations are part of the manuscripts’ mathematical claims, reported here without proof verification.

For M1, established low-degree nonvanishing results still detect its class under maximal assembly. The class is lost on passing to the reduced completion. Consequently the claimed reduced rational-injectivity failure does not disprove maximal rational injectivity or classical Novikov. Terminology such as “strong Novikov” varies, so specifying the completion is essential.

Likewise, projections in a von Neumann algebra, matrices over a reduced C*-algebra, the reduced algebra itself, and the algebraic group ring have different implications. The papers carefully move between these settings only when an additional construction permits it. Understanding which geometric restrictions restore assembly or projectionlessness remains meaningful even if the unrestricted conjectures fail.

8. Sources and further reading

• Higson, Lafforgue and Skandalis, Counterexamples to the Baum–Connes conjecture, for the earlier coefficient and groupoid obstructions.
• M1’s historical discussion identifies the low-degree maximal-assembly nonvanishing theorem and the corrected exotic-completion statement used to delimit its conclusion.
• M2 and M3 explain why a uniform spectral gap is needed in addition to a nonintegral or irrational von Neumann dimension; their bibliographies give the original projection, trace and expander references.

Source manuscripts

• M1. OpenAI, A torsion-free counterexample to reduced Baum–Connes injectivity (September 23, 2026).
• M2. OpenAI, A torsion-free counterexample to the Kadison–Kaplansky projection conjecture (September 23, 2026).
• M3. OpenAI, An irrational-trace counterexample to reduced Baum–Connes (September 23, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.