Understand coideals in property (T) discrete quantum groups from bicrossed products

Investigate and characterize the structure of coideals in property (T) discrete quantum groups arising from compact bicrossed products, including identifying and describing finite-dimensional coideals and their properties relevant to quantum Schreier graph constructions.

Background

Coideals play the role of quantum homogeneous spaces and are central to defining quantum Schreier graphs. For using bicrossed product examples in expander constructions, understanding the coideal structure is crucial.

Even in the bicrossed product setting—where property (T) examples are available—the structure of coideals is insufficiently understood, limiting explicit constructions of quantum Schreier graph expanders.