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2018

A group with at least subexponential hyperlinear profile

Slofstra, William

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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.

A group with at least subexponential hyperlinear profile · Around

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G. Higman, Subgroups of finitely presented groups , Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 262
1961
Earlier work this paper cites.
D. Kazhdan, On ϵ \epsilon -representations , Israel Journal of Mathematics 43
1982
Earlier work this paper cites.
J.-C. Birget, A. Yu Ol’shanskii, E. Rips, and M. V. Sapir, Isoperimetric Functions of Groups and Computational Complexity of the Word Problem , Annals of Mathematics 156
2002
Earlier work this paper cites.

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