Classify property (T) for duals of compact bicrossed product quantum groups

Determine necessary and sufficient conditions, in terms of the matched pair data (Γ, G, α, β), under which the discrete dual \hat{\mathbb G} of the compact bicrossed product quantum group \mathbb G(Γ, G, α, β) has Kazhdan’s property (T), thereby providing a complete classification in terms of the input data.

Background

Bicrossed products produce compact quantum groups \mathbb G(Γ, G, α, β) from a matched pair consisting of a discrete group Γ and a compact group G with compatible actions α and β. Their discrete duals \hat{\mathbb G} can, in many cases, have property (T), with some sufficient conditions known (e.g., when G is finite and Γ has property (T)).

However, a general characterization linking the matched pair data directly to property (T) for \hat{\mathbb G} is currently unavailable. A full classification would identify exactly which inputs (Γ, G, α, β) yield property (T) on the dual side.