Unitary easy quantum groups: Geometric aspects

We discuss the classification problem for the unitary easy quantum groups, under strong axioms, of noncommutative geometric nature. Our main results concern the intermediate easy quantum groups ON⊂G⊂UN+. To any such quantum group we associate its Schur-Weyl twist Ḡ, two noncommutative spheres S,S̄, a noncommutative torus T, and a quantum reflection group K. Studying (S,S̄,T,K,G,Ḡ) leads then to some natural axioms, which can be used in order to investigate G itself. We prove that the main examples are covered by our formalism, and we conjecture that in what concerns the case UN⊂G⊂UN+, our axioms should restrict the list of known examples.