Paths in quantum Cayley trees and $L^2$-cohomology

Roland Vergnioux (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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 $A_o(I_n)$.

Domaines: Algèbres d'opérateurs [math.OA]

Roland Vergnioux. Paths in quantum Cayley trees and $L^2$-cohomology. Advances in Mathematics, 2012, 229 (5), pp.2686-2711. ⟨10.1016/j.aim.2012.01.011⟩. ⟨hal-00682500⟩