A group with at least subexponential hyperlinear profile

The hyperlinear profile of a group measures the growth rate of the dimension of unitary approximations to the group. We construct a finitely-presented group whose hyperlinear profile is at least subexponential, i.e. at least $\exp(1/\epsilon^{k})$ for some $0 < k < 1$. We use this group to give an example of a two-player non-local game requiring subexponential Hilbert space dimension to play near-perfectly.