Existence of non-hyperlinear groups

Determine whether non-hyperlinear groups exist, i.e., construct or rule out groups that admit no asymptotically faithful unitary actions on finite-dimensional Hilbert spaces.

Background

The pursuit of perfect completeness in LIN-MIP* connects to a long-standing question in group theory: the existence of non-hyperlinear groups, which are groups without asymptotically faithful unitary representations on finite-dimensional Hilbert spaces. The paper notes that showing LIN-MIP*_{1,s}=RE for some constant s<1 would imply the existence of such groups, underscoring the significance of this open question beyond complexity theory.

The authors highlight this question as a motivating backdrop to their results, reflecting its importance in the broader landscape linking interactive proofs, operator algebras, and geometric group theory.