Entanglement in Non-local Games and the Hyperlinear Profile of Groups

We relate the amount of entanglement required to play linear system non-local games near-optimally to the hyperlinear profile of finitely presented groups. By calculating the hyperlinear profile of a certain group, we give an example of a finite non-local game for which the amount of entanglement required to play -optimally is at least O(1/ k), for some k > 0. Since this function approaches infinity as approaches zero, this provides a quantitative version of a theorem of the first author.