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Math-2835 · mathematics · Analysis

Stable recovery of twisted lattice factors

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For an ICC group $\Gamma $ in the class of groups abstractly commensurable with lattices in a finite nonempty product $H_i(k_i)^+$, where $k_i$ is R, C or a finite extension of $Q_p$ (with primes allowed to differ), each $H_i$ is a connected adjoint absolutely simple algebraic group, $H_i(k_i)^+$ is generated by the $k_i-points$ of unipotent radicals of proper parabolic subgroups, and each such topological group is noncompact and has property (T), an arbitrary countably infinite ICC group $\Lambda $, normalized scalar cocycles $\mu ,\omega $ and positive scales t,s, does $L_\mu (\Gamma )^t\cong L_\omega (\Lambda )^s$ hold exactly when $t=s$ and a group isomorphism $\Gamma \to \Lambda $ carries $\omega $ to $\mu $ up to a scalar coboundary, with every specified factor isomorphism implemented by that group/cochain map and an inner corner equivalence?

Short English statement

For an ICC group $\Gamma $ in the class of groups abstractly commensurable with lattices in a finite nonempty product $H_i(k_i)^+$, where $k_i$ is R, C or a finite extension of $Q_p$ (with primes allowed to differ), each $H_i$ is a connected adjoint absolutely simple algebraic group, $H_i(k_i)^+$ is generated by the $k_i-points$ of unipotent radicals of proper parabolic subgroups, and each such topological group is noncompact and has property (T), an arbitrary countably infinite ICC group $\Lambda $, normalized scalar cocycles $\mu ,\omega $ and positive scales t,s, does $L_\mu (\Gamma )^t\cong L_\omega (\Lambda )^s$ hold exactly when $t=s$ and a group isomorphism $\Gamma \to \Lambda $ carries $\omega $ to $\mu $ up to a scalar coboundary, with every specified factor isomorphism implemented by that group/cochain map and an inner corner equivalence?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2835-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2835

Manuscript 1 (family 286, 2026-09-23): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Stable recovery of twisted lattice factors

OpenAI's claim

Stable isomorphisms of arbitrarily twisted factors from the specified property-T lattice class are claimed to recover the group and cocycle, preserve amplification scales, and fit an explicit arithmetic correspondence category. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $\mathscr K$ consist of ICC groups that are countably infinite and abstractly commensurable with lattices in products, with a positive finite number of factors, of $\mathbf H_i(k_i)^+$, where $k_i$ is real, complex or a finite extension of a $p$-adic field, $\mathbf H_i$ is connected adjoint absolutely simple, and each factor is noncompact with property $(T)$. The plus group is generated by rational points of unipotent radicals of proper parabolics; ICC means all nonidentity conjugacy classes are infinite, and abstract commensurability means isomorphic finite-index subgroups. No irreducibility, cocompactness or torsion premise is imposed. For $\Lambda\in\mathscr K$, any countably infinite ICC $\Pi$, normalized scalar cocycles $\omega,\nu$ and positive amplification scales $t,s$, OpenAI claims \[ L_\omega(\Lambda)^t\simeq L_\nu(\Pi)^s\quad\Longleftrightarrow\quad t=s\text{ and }\exists\delta:\Lambda\simeq\Pi,\ b:\Lambda\to S^1:\ \omega(g,h)=b(g)b(h)b(gh)^{-1}\nu(\delta g,\delta h). \] Amplification $Q^t$ is a matrix corner of unnormalized trace $t$. Every specified unit-scale isomorphism is claimed to have $u_g\mapsto wb(g)v_{\delta g}w^*$ for a target unitary $w$. At matrix corners it is implemented by a rectangular partial isometry after the canonical entrywise map, with the initial and final corner projections. It also claims the recovered comparison group belongs to $\mathscr K$ and every such factor has fundamental group $\{1\}$.

Outline of the argument

Manuscript 1 (family 286, 2026-09-23): Arithmetic exhaustion reduces finite correspondences to subgroup models, whose actual summands have integral module dimensions. An equivalence correspondence has reciprocal dimensions, forcing both to equal one and recovering the prescribed stable isomorphism canonically. Homogeneous multiplication and minimal-corner arguments then reconstruct finite-index neighbors, while projective induction computes the relevant fusion and cocycle-ratio data.

Formal verification

Family 286 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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