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OpenAI reports arithmetic classification of twisted lattice-factor correspondences

Individual problem

For an ICC group commensurable with a lattice in a finite product of the specified property-(T) algebraic groups over characteristic-zero local fields, any countably infinite ICC comparison group, and arbitrary normalized scalar cocycles at both ends, every bifinite correspondence between their twisted group factors is a closed bimodule summand of a finite sum of models from finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio.

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OpenAI reports arithmetic classification of twisted lattice-factor correspondences

  • Mathematics

For an ICC group commensurable with a lattice in a finite product of the specified property-(T) algebraic groups over characteristic-zero local fields, any countably infinite ICC comparison group, and arbitrary normalized scalar cocycles at both ends, every bifinite correspondence between their twisted group factors is a closed bimodule summand of a finite sum of models from finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio. Conversely, every such model, sum, and closed summand is bifinite.

AI involvementAI-led
ImportanceNotable
Technical verificationReported
Human understandingFew

Understanding the problem

Read the background

For an ICC group commensurable with a lattice in a finite product of the specified property-(T) algebraic groups over characteristic-zero local fields, any countably infinite ICC comparison group, and arbitrary normalized scalar cocycles at both ends, every bifinite correspondence between their twisted group factors is a closed bimodule summand of a finite sum of models from finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio. Conversely, every such model, sum, and closed summand is bifinite.

Why it matters

If correct, identifies the arithmetic source of every finite correspondence in a broad property-(T) lattice class despite arbitrary scalar twists and an unrestricted ICC comparison group.

What counts as progress

Do bifinite correspondences of these twisted lattice group factors come from finite-index subgroup isomorphisms?

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