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The arithmetic category and stable recovery of lattice factors
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Abstract
We prove stable canonical recovery for scalar-twisted group factors of ICC groups commensurable with property-$(T)$ lattices over characteristic-zero local fields. Every specified stable isomorphism to a scalar-twisted factor of any countably infinite ICC group recovers the group, the cocycle up to a normalized cochain, and equal amplification scales, with an implementing partial isometry in the given matrix corners. We compute the arithmetic correspondence subcategory for arbitrary countable ICC groups, including all bounded maps, summands, conjugates, and Connes fusion, and combine it with the companion's arithmetic exhaustion theorem. Multiplication also recovers arbitrary finite-index factor neighbors, with the exact projective condition on the matched cocycle ratio.
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Comments and discussion
- Cyril Houdayer, reaction in 100+ reactions to 100+ solutions (Proofs and Prompts, 8 October 2026; collection updated 19 October 2026). Discusses the announced lattice case of Connes’ rigidity conjecture in relation to his work with Rémi Boutonnet on noncommutative boundary theory and describes plans to study the proof with students and postdocs, without reporting a completed verification. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Cyril Houdayer, Operator algebraic characterization of the noncommutative Poisson boundary (arXiv:2410.11707, C. R. Math. Acad. Sci. Paris 363 (2025), 199–204). Presents a boundary characterization as further evidence towards Connes' rigidity conjecture for higher-rank lattices, prior partial evidence toward the stable canonical recovery statement in Theorem 1.2; not in the paper's bibliography.
- Ionut Chifan, Junhwi Lim, Property (T) group factors whose Jones index set equals all positive integers (arXiv:2511.04822, November 2025). Constructs a continuum of pairwise non-stably-isomorphic property (T) group factors whose Jones index set is exactly the positive integers, which compares with the paper's claim that bifinite correspondences of lattice factors have positive integer dimensions and with its finite-index factor neighbor classification.
- Uri Bader, Rémi Boutonnet, Cyril Houdayer, Jesse Peterson, Charmenability of arithmetic groups of product type (arXiv:2009.09952, Invent. Math. 229 (2022), 929–985). Studies characters and normal ucp maps for arithmetic groups of product type, operator-algebraic rigidity context for the product-type lattices in the class 𝒦 to which the recovery theorems apply; not in the paper's bibliography.
- Cyril Houdayer, Noncommutative rigidity of higher rank lattices (IHP lecture notes, lectures given June 2024). Surveys noncommutative boundary structures and their connection to Connes’ rigidity conjecture, providing background for the companion’s geometric exhaustion input to this paper’s claimed stable canonical recovery theorem. [family-level]
Elsewhere
- Emergent Mind: family 286
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026,
author = {{OpenAI}},
title = {{The arithmetic category and stable recovery of lattice factors}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026/paper.pdf}{OAI:The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026}},
year = {2026}
}Family 286: Rigidity and arithmetic of lattice von Neumann algebras
Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-$(T)$ lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such bimodule is a summand of finite sums of models arising from finite-index subgroup isomorphisms and finite-dimensional projective representations. The classification also recovers the group, scalar cocycle and amplification scale up to the stated stable equivalence.
Other papers in this family:
- Arithmeticity of twisted finite correspondences for lattices over local fields · September 23, 2026