Classify groups that admit strongly convergent representations

Determine which finitely generated groups G admit sequences of random unitary representations ρ_N:G→U(D_N) (or random permutation representations) that converge strongly to the regular representation λ_G, in the sense that for every x∈ℂ[ G ], lim_{N→∞} ||ρ_N(x)|| = ||λ_G(x)|| in probability.

Background

The survey describes strong convergence for free groups, limit groups (via random permutation representations), and right-angled Artin groups (via unitary representations), along with notable obstructions (e.g., for SL_4(ℤ)).

Despite significant progress, the overall classification problem—identifying all groups that support strongly convergent representation models—remains unresolved.