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Math-2320 · mathematics · Topology

Baum–Connes Conjecture for Torsion-Free Groups

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a torsion-free discrete group $G$, the Baum–Connes assembly map from $K$-homology $K_*(BG)$ to $K_*(C_r^*(G))$ should be an isomorphism.

Formulation references: [S1], [S2].

Short English statement

For a torsion-free discrete group $G$, the Baum–Connes assembly map from $K$-homology $K_*(BG)$ to $K_*(C_r^*(G))$ should be an isomorphism.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2320-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2320

Manuscript 1 (family 285, 2026-09-23): The infinite-order degree-zero assembly-kernel class for a finitely generated torsion-free discrete group directly refutes the universal coefficient-free reduced injectivity and isomorphism assertions, with no coefficient or maximal-completion substitution.

Manuscript 2 (family 285, 2026-09-23): The nonintegral scalar projection class lies outside the coefficient-free reduced assembly image, so surjectivity and hence the universal isomorphism assertion fail; torsion freeness is used only in the projection-conjecture paper.

Baum–Connes Conjecture for Torsion-Free Groups

OpenAI's claim

Manuscript 1 (family 285, 2026-09-23): For one finitely generated group without torsion, the coefficient-free reduced assembly map annihilates an infinite-order degree-zero K-homology class; this is claimed to refute rational injectivity. Manuscript 2 (family 285, 2026-09-23): OpenAI claims the existence of a nontrivial scalar projection inside the reduced C*-algebra of a finitely generated torsion-free group, contradicting the Kadison--Kaplansky assertion.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims a discrete group, torsion-free and finitely generated, $G$, an injection $i:\mathbb Z^2\hookrightarrow G$, and $c\in H^2(BG;\mathbb Z)$ such that \[ \langle(Bi)^*c,[T^2]\rangle=1,\qquad i_*\beta=0\text{ in }K_0(C_r^*(G)), \] here $\beta\in K_0(C_r^*(\mathbb Z^2))$ is the Bott generator chosen to have rank zero. The class $h=(Bi)_*[D_{T^2}]\in K_0(BG)$ has infinite order but satisfies $\mu_G^r(h)=0$. Here $\mu_G^r:K_*(BG)\to K_*(C_r^*(G))$ is reduced coefficient-free analytic assembly and $[D_{T^2}]$ is the oriented torus spin-Dirac class. This asserts failure of reduced assembly injectivity without coefficients, even rationally.
Mathematical claim 2.
OpenAI claims a discrete group, torsion-free and finitely generated, $G$ whose reduced group $C^*$-algebra contains a scalar projection $e=e^*=e^2$ with \[ 0<\tau_G(e)<\tfrac12,\qquad\tau_G(x)=\langle x\delta_1,\delta_1\rangle. \] Thus $e$ is neither $0$ nor $1$. This concerns a scalar projection in the operator-norm completion of the left regular group algebra, and supplies a counterexample to the torsion-free projection assertion. The manuscript also uses the integral trace restriction on reduced assembly for torsion-free groups to place this projection class outside the assembly image.

Outline of the argument

Manuscript 1 (family 285, 2026-09-23): The construction embeds a torus subgroup with an integral cohomology detector while arranging disappearance of its Bott class analytically. Operators assembled on almost disjoint Cayley-graph sheets retain controlled spectral supports; estimates for the leftover forest permit the required homotopies in the reduced completion. A finite witness then restricts the construction to a finitely generated subgroup.

Manuscript 2 (family 285, 2026-09-23): A magnetic tower model produces a separated spectral band carrying a matrix projection of positive trace below one half. Graph sampling and small-cancellation recovery preserve this band after the model is represented on the group, with intersection losses bounded by trace estimates. A free-product compression converts the matrix projection into a scalar one while retaining torsion-freeness.

Formal verification

Family 285 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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