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.

Manuscripts (2)

Arithmeticity of twisted finite correspondences for lattices over local fields

September 23, 2026 · source · cite as \cite{OAI:Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-September-23-2026}

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.

The arithmetic category and stable recovery of lattice factors

September 23, 2026 · source · cite as \cite{OAI:The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026}

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.

Result 286 · Operator algebras. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.