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Arithmeticity of twisted finite correspondences for lattices over local fields
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Abstract
We prove arithmetic exhaustion for bifinite correspondences between arbitrarily scalar-twisted group factors of ICC groups commensurable with property-$(T)$ lattices over characteristic-zero local fields and twisted group factors of arbitrary countable ICC groups. Every such correspondence is a closed summand of a finite direct sum of models obtained from actual finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio. The proof includes reducible products, mixed real and finite places, and quaternionic and Cayley rank-one factors.
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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)
- 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 and groups acting on products of trees, operator-algebraic rigidity input in the product/mixed-place setting covered by the paper's Definition 1.1; not in the paper's bibliography.
- Alon Dogon, Michael Glasner, Yuval Gorfine, Liam Hanany, Arie Levit, Non-uniform higher-rank lattices are character rigid (arXiv:2507.21862, July 2025). Proves character rigidity for all non-uniform irreducible higher-rank lattices, relating to the character-rigidity precedents surveyed in the introduction and to the paper's explicit inclusion of nonuniform lattices.
- Cyril Houdayer, Operator algebraic characterization of the noncommutative Poisson boundary (arXiv:2410.11707, C. R. Math. Acad. Sci. Paris 363 (2025), 199–204). Characterizes the boundary inclusion L(Γ) ⊂ L(Γ ↷ B) as further evidence towards Connes' rigidity conjecture for higher-rank lattices, giving boundary-method context for the paper's boundary-measurement strategy; 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, the Nevo–Zimmer structure theorem and a noncommutative Margulis factor theorem, providing background for the boundary-measurement approach to the claimed arithmetic exhaustion theorem. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 286
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-September-23-2026,
author = {{OpenAI}},
title = {{Arithmeticity of twisted finite correspondences for lattices over local fields}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-September-23-2026/paper.pdf}{OAI:Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-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:
- The arithmetic category and stable recovery of lattice factors · September 23, 2026