Hyperlinear

Hyperlinear is a property of discrete groups, most often studied in the context of operator algebras and group theory. A countable group G is hyperlinear if its structure can be approximated by finite-dimensional unitary groups in a precise metric sense, or, equivalently, if its group von Neumann algebra L(G) embeds into an ultrapower of the hyperfinite II1 factor.

One formulation uses almost-homomorphisms into unitary groups. G is hyperlinear if there exists a sequence of maps φn: G → U(n) into the n-by-n unitary group such that

- φn(1) is the identity for all n,

- for every finite subset F ⊂ G and ε > 0, and for all g,h in F with gh in F, the normalized Hilbert–Schmidt distance dHS(φn(g)φn(h), φn(gh)) goes to 0 as n → ∞,

- and the maps are asymptotically faithful on F, meaning nontrivial elements are not sent arbitrarily close to the identity in the limit.

Equivalently, G embeds into an ultraproduct ∏ω U(ni) of finite-dimensional unitary groups with respect to the normalized Hilbert–Schmidt metric.

A key equivalence is that G is hyperlinear if and only if the group von Neumann algebra L(G) is Connes-embeddable, i.e., embeds into the ultrapower Rω of the hyperfinite II1 factor.

Examples and relations: All finite groups and all amenable groups are hyperlinear. The class is closed under taking subgroups and direct products. It is unknown whether every countable group is hyperlinear, and the broader question is closely tied to Connes’ embedding problem. Hyperlinearity is part of a family of approximation properties that includes soficity; the precise relationships between these classes remain active areas of research.