A hyperlinear group is a group whose finite portions admit arbitrarily accurate unitary matrix
models in normalized Hilbert-Schmidt norm
that keep distinct elements separated (Pestov 2008). For an 
complex matrix 
, this norm is
 |
(1)
|
where 
is the conjugate transpose and 
denotes the matrix trace
of a matrix 
. Precisely, for every finite
symmetric subset 
containing the identity element 
and every 
, there are a positive
integer 
and a map 
with 
such that
 |
(2)
|
whenever 
, and
 |
(3)
|
for distinct 
. Here 
is the unitary group,
is the identity matrix, and the matrix
size is unrestricted. Every sofic group is hyperlinear.
This definition differs from the operator norm approximation
defining an MF group.
For a countable group 
, hyperlinearity is equivalent to a trace-preserving
embedding of its group von Neumann algebra 
, with its canonical trace, into an ultrapower
of the hyperfinite type 
factor (Pestov 2008). A nonembeddable tracial algebra with
some other trace does not by itself give a nonhyperlinear group.
Alekseev et al. (2026) reported a proof of the existence of nonhyperlinear groups. More generally, let 
be groups with Kazhdan's
property (T) such that the elements 
satisfying 
generate 
. If 
is not normal in 
, then the amalgamated
free product 
is not hyperlinear. Their proof builds on Thom's earlier
conditional argument. The work builds on a proof obtained
by Danus using GPT-6 Astra, and the joint proof was developed
with mathematical assistance from ChatGPT and Codex.
