  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).
