Paths in quantum Cayley trees and L^2-cohomology

Abstract

We study existence, uniqueness and triviality of path cocycles in the quantum Cayley graph of universal discrete quantum groups. In the orthogonal case we find that the unique path cocycle is trivial, in contrast with the case of free groups where it is proper. In the unitary case it is neither bounded nor proper. From this geometrical result we deduce the vanishing of the first L^2-Betti number of A_o(I_n).

30 pages ; v2: major update with many improvements and new results about the unitary case ; v3: accepted version

References in corpus (4)

• The Baum-Connes conjecture for free orthogonal quantum groups
• The Property of Rapid Decay for Discrete Quantum Groups
• Orientation of quantum Cayley trees and applications
• Homology of free quantum groups

Cited by in corpus (3)

• Orthogonal free quantum group factors are strongly 1-bounded
• L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups
• Measure continuous derivations on von Neumann algebras and applications to L^2-cohomology