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Math-2357 · mathematics · Group theory

Rigidity and Assembly Conjectures for Discrete Groups

Status Partially solved · unsolvedmath.comLLM status Open unverified

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 group $G$ and integral domain $R$, are $0$ and $1$ the only idempotents of $RG$? Do the negative K-groups of $\mathbb{Z}G$, the reduced group $K_0(\mathbb{Z}G)$, and $\operatorname{Wh}(G)$ vanish? Are higher signatures homotopy invariant, closed aspherical manifolds topologically rigid, and groups of type FP of type FF? Is every finitely presented Poincaré-duality group of dimension at least five the fundamental group of an aspherical homology ANR-manifold? Is every hyperbolic group with boundary $S^n$ the fundamental group of a closed aspherical manifold?

Formulation references: [S1], [S2].

Short English statement

For a torsion-free group $G$ and integral domain $R$, are $0$ and $1$ the only idempotents of $RG$? Do the negative K-groups of $\mathbb{Z}G$, the reduced group $K_0(\mathbb{Z}G)$, and $\operatorname{Wh}(G)$ vanish? Are higher signatures homotopy invariant, closed aspherical manifolds topologically rigid, and groups of type FP of type FF? Is every finitely presented Poincaré-duality group of dimension at least five the fundamental group of an aspherical homology ANR-manifold? Is every hyperbolic group with boundary $S^n$ the fundamental group of a closed aspherical manifold?

Sources

Catalogue entry

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

Contributions 1

Math-2357-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2357

Corollary1.2(ii) proves the idempotent assertion for characteristic-zero domains, but leaves positive characteristic and the independent rigidity and assembly assertions unanswered. The main theorem and its closed-hyperbolic-three-manifold consequences settle the S^2-boundary subcase. The record quantifies over additional boundary types, general hyperbolic groups, dimensions, or independent bundle clauses that this paper does not settle. The nonhomeomorphic aspherical pair refutes the Borel-rigidity clause of this broad bundle, but does not decide its idempotent, K-group, signature, FP/FF or higher-dimensional realization clauses. This entry records partial support; complete coverage of the question is not established by this catalogue.

Rigidity and Assembly Conjectures for Discrete Groups

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: The Bass trace restriction is claimed for complex group rings of arbitrary discrete groups, implying triviality of scalar idempotents for torsion-free groups over characteristic-zero domains; A hyperbolic group with two-sphere boundary is claimed to act geometrically on hyperbolic three-space with finite kernel; Two homotopy-equivalent closed aspherical four-manifolds with hyperbolic fundamental group are claimed nonhomeomorphic, with an additional nongeometric self-homotopy equivalence. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Manuscript 1.
For an arbitrary discrete group $G$, let $\operatorname{HS}_G$ be the Hattori--Stallings trace, whose value at a conjugacy class sums the coefficients on that class. For every $x\in K_0(\mathbb C[G])$, \[ |g|=\infty\quad\Longrightarrow\quad\operatorname{HS}_G(x)([g])=0. \] The torsion-free corollary states that, for every $q\ge1$ and idempotent $e\in M_q(\mathbb C[G])$, its identity-coefficient trace equals the rank of its augmentation matrix: \[ \tau_{G,q}(e)=\operatorname{rank}_{\mathbb C}(\epsilon_{G,q}(e)). \] For such $G$, any commutative unital domain $R$ of characteristic zero satisfies \[ e\in R[G],\ e^2=e\quad\Longrightarrow\quad e\in\{0,1\}. \] The characteristic-zero qualification does not cover positive-characteristic group rings.
Manuscript 2.
Let $G$ be word-hyperbolic with Gromov boundary $\partial G\cong S^2$. The source claims a homomorphism \[ \rho:G\longrightarrow\operatorname{Isom}(\mathbb H^3) \] whose action on hyperbolic three-space is proper and cocompact, and whose kernel is finite. With the additional assumption that $G$ is torsion-free, the corollaries identify $G\cong\pi_1(M)$, where $M$ is closed and has a hyperbolic metric in dimension three, and supply a finite-index subgroup \[ H\cong\pi_1(\Sigma)\rtimes\mathbb Z, \] where $\Sigma$ is a closed connected orientable surface. They also assert residual finiteness, virtual compact specialness, and positive virtual first Betti number in that torsion-free case.
Manuscript 3.
The construction claims two closed connected aspherical topological four-manifolds $M,N$ with \[ M\simeq N,\qquad M\not\cong N\text{ by homeomorphism},\qquad \pi_1M\cong\pi_1N\text{ word-hyperbolic}. \] Thus homotopy equivalence and hyperbolicity do not force topological homeomorphism for this claimed four-dimensional pair.

Outline of the argument

An idempotent is encoded by finite cyclic path configurations whose weighted deletion relations generate cycles. Exterior differential forms and a cyclic Pfaffian identity detect forbidden conjugacy support; trace-rank arguments then turn the complex trace conclusion into the stated idempotent consequence.

The argument applies a boundary modulus criterion. Unbounded modulus produces scalar limits analyzed by exponential records or subexponential probability flows; comparing same-depth expansions forces the required measure control and the three-dimensional hyperbolic action.

Reflected cyclic covers realize the relevant symmetries while preserving asphericity. A parity-chamber construction and cyclic-cover criteria distinguish manifold models despite the common homotopy type, and the realized symmetry gives the self-equivalence obstruction.

Formal verification

For family 207, the documented Lean statements give complex group-ring Bass trace support and characteristic-zero torsion-free trace rank and idempotent triviality. The separate $\ell^1$ theorem is not selected. The listed Comparator files are BassTorsionFree.lean, BassTrace.lean. Named declarations include OAI.BassTrace.RightProjective.bassTraceModules_vanishing_and_support. For family 246, the documented theorem gives a proper cocompact hyperbolic-three-space action with finite kernel from the two-sphere boundary hypothesis. The listed Comparator files are CannonGeometricAction.lean. The inventory does not supply a named declaration for these Comparator entries. Family 320 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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