Paths in quantum Cayley trees and L^2-cohomology

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

2009-05-11

arxiv.org/abs/0905.1732v3

Adv. Math. 229 (2012) 2686-2711

Mathematics

Operator Algebras

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

Scientific paper

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).

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