MF Group

An MF group is a countable discrete group that embeds in the group of unitary elements of a C^*-algebra of the form

where denotes the algebra of complex matrices. The product consists of sequences bounded in operator norm, and the denominator consists of sequences whose operator norms tend to zero (Carrión et al. 2013). Thus finite portions of the group operation admit increasingly accurate unitary matrix models in operator norm. Every countable residually finite group is MF.

Eckhardt (2026) reports an explicit construction of groups that are not MF, crediting the mathematical ideas and proofs to ChatGPT 5.6 Sol. Take the semidirect product and , and form the ascending HNN extension . Then the generalized wreath product

is not MF. Here acts on the left cosets by left translation, and the direct sum consists of finitely supported lamp configurations. The obstruction uses Kazhdan's property (T) of .

The operator norm approximation in this definition is different from the normalized Hilbert-Schmidt norm approximation defining a hyperlinear group. An embedding of the reduced group C^*-algebra in implies the group property, but the definition itself prescribes neither such an algebra embedding nor a canonical trace.