we-solve (beta)

Math-2834 · mathematics · Analysis

Arithmetic classification of twisted lattice correspondences

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let $\Gamma $ be a countably infinite ICC group commensurable with a property-(T) lattice 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), and $\Lambda $ any countably infinite ICC group. For arbitrary normalized circle-valued cocycles $\mu $ and $\omega $, is every bifinite $L_\mu (\Gamma )–L_\omega (\Lambda )$ correspondence a closed summand of finitely many models obtained by induction from actual isomorphisms between finite-index subgroups and finite-dimensional projective representations with multiplier $\mu /\delta *\omega $?

Short English statement

Let $\Gamma $ be a countably infinite ICC group commensurable with a property-(T) lattice 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), and $\Lambda $ any countably infinite ICC group. For arbitrary normalized circle-valued cocycles $\mu $ and $\omega $, is every bifinite $L_\mu (\Gamma )–L_\omega (\Lambda )$ correspondence a closed summand of finitely many models obtained by induction from actual isomorphisms between finite-index subgroups and finite-dimensional projective representations with multiplier $\mu /\delta *\omega $?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2834-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2834

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.

Arithmetic classification of twisted lattice correspondences

OpenAI's claim

Finite correspondences involving arbitrarily twisted ICC lattice factors over characteristic-zero local fields are claimed to arise from finite-index group identifications and finite-dimensional projective fibers, including mixed-place and eligible rank-one lattices. 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 $\Gamma\in\mathscr K$, any countably infinite ICC $\Lambda$ and normalized scalar cocycles $\mu,\omega$, OpenAI claims every separable bifinite $L_\mu(\Gamma)$--$L_\omega(\Lambda)$ correspondence is a closed bimodule summand of a finite sum of induced models \[ E=L^2(L_\mu(\Gamma))\ \overline\otimes_{L_{\mu|_A}(A)}\ (V_\sigma\otimes L^2(L_\omega(\Lambda))), \] for finite-index $A\le\Gamma$, $B\le\Lambda$, actual isomorphism $\delta:A\to B$ and finite-dimensional projective unitary $\sigma$ with \[ \sigma(a)\sigma(b)=\frac{\mu(a,b)}{\omega(\delta(a),\delta(b))}\sigma(ab). \] Conversely these models and summands are bifinite. For $n=\dim V_\sigma$ it claims left dimension $n[\Lambda:B]$ and right dimension $n[\Gamma:A]$. Twisted factors have canonical unitaries multiplying by the displayed cocycles; bifinite means finite module dimension on both sides, not mutual-commutant actions.

Outline of the argument

Manuscript 1 (family 286, 2026-09-23): The argument removes compact projective pieces and constructs boundary observations within the remaining weakly mixing correspondence. Positivity is upgraded to commuting projection-valued measurements, allowing their joint distribution to recover finite-dimensional pieces permuted by both groups. Stabilizers of these pieces supply actual finite-index subgroup isomorphisms, with cocycle ratios recorded by the projective fibers.

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

Go to contribution →