Paths in quantum Cayley trees and L^2-cohomology

TL;DR

This paper investigates path cocycles in quantum Cayley graphs of universal discrete quantum groups, revealing their triviality or boundedness and deriving implications for L^2-cohomology.

Contribution

It provides new insights into the properties of path cocycles in quantum Cayley trees and their impact on the first L^2-Betti number of certain quantum groups.

Findings

Unique path cocycle in orthogonal case is trivial

Path cocycle in unitary case is neither bounded nor proper

First L^2-Betti number of A_o(I_n) vanishes

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).