Draft:MF group

• Comment: Fails WP:INDISCRIMINATE and lacks context, it's not understandable to a lay reader what the importance of this topic is. Simply because something exists does not make it deserving of an independent article. There is also very little content in it. ᴢxᴄᴠʙɴᴍ (ᴛ) 22:54, 30 August 2026 (UTC)

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In mathematics, an MF group (standing for matricial field group) is a countable discrete group which can be approximated by finite-dimensional unitary groups using the operator norm.

Informally, the MF condition asks whether finite parts of a group's multiplication can be modelled by matrices with errors as small as desired. The matrices approximately maintain multiplication while also keeping distinct elements from acting the same. The finite matrices can be any size, or increase in size as smaller errors are required.

MF groups are studied in group theory and operator algebras as part of the study of finite-dimensional approximations of infinite groups. The subject connects questions about groups with approximation properties of their associated C*-algebras and with the problem of correcting approximate representations to exact ones.

Definition

A group $\Gamma$ is MF if it embeds into $$U\left({\frac {\prod _{k}M_{n_{k}}(\mathbb {C} )}{\bigoplus _{k}M_{n_{k}}(\mathbb {C} )}}\right)$$ for a sequence of positive integers $(n_{k})$, where $M_{n}(\mathbb {C} )$ is the algebra of $n\times n$ complex matrices, and sequences whose operator norms tend to zero compose the denominator.

The product in this expression consists of uniformly norm-bounded matrix sequences. Passing to the quotient identifies two such sequences when the operator norm of their difference tends to zero.

Also, finite parts of the group allow for maps to unitary matrices which are approximately multiplicative and which keep group elements a fixed distance apart in operator norm.

Equivalently, there are positive integers $d_{k}$ and maps $\varphi _{k}:\Gamma \to U(d_{k})$ such that $$\lim _{k\to \infty }\|\varphi _{k}(gh)-\varphi _{k}(g)\varphi _{k}(h)\|_{\mathrm {op} }=0\qquad (g,h\in \Gamma ),$$ and $$\liminf _{k\to \infty }\|\varphi _{k}(g)-I_{d_{k}}\|_{\mathrm {op} }>0\qquad (g\neq e).$$ Here $e$ is the identity element of the group and $I_{d_{k}}$ is the identity matrix. The first condition makes multiplication errors disappear in the limit; the second prevents a nonidentity element from becoming indistinguishable from the identity.

The operator norm of a matrix $A$ is $\sup _{\|v\|=1}\|Av\|$. Thus, it measures the largest error over all unit vectors. The approximation conditions concern each fixed collection of group elements; they do not require one error bound to hold simultaneously over the entire group.

Examples and properties

All groups that are locally embeddable into finite groups are MF. All free groups are also MF. A group being MF is the same as the group C*-algebra having quasidiagonality, for amenable groups. Every amenable group is MF.

For a LEF group, each finite portion of its multiplication table can be reproduced exactly in a finite group. Representing these finite groups by permutation matrices yields the required unitary approximations.

MF approximation also extends beyond LEF groups. Carrión, Dadarlat and Eckhardt proved that the quotient of a countable residually finite group by a central subgroup is MF. They applied this to a quotient of an Abels group that is finitely presented and solvable but is not LEF. Other classes include countable linear groups and groups locally embeddable in amenable groups.

Operator algebras and stability

Carrión, Dadarlat and Eckhardt introduced their group definition in analogy with MF C*-algebras, which embed into quotients of products of matrix algebras of the same form. The result of Tikuisis, White and Winter on amenable groups settled Rosenberg's conjecture that discrete amenable groups have quasidiagonal C*-algebras.

A related research question is whether approximate unitary representations can be perturbed to exact representations. This is a question of stability: the existence of approximations and the possibility of correcting them are separate properties. For example, $\mathbb {Z} ^{2}$ is MF, but there are pairs of almost commuting unitary matrices that cannot be approximated by commuting pairs. Dadarlat's survey discusses how group cohomology supplies obstructions to stability, including results for MF groups.

Related approximation properties and terminology

Several group approximation properties use the same general framework with different finite models or metrics.

Property	Finite models	Measurement of error
Soficity	Permutations of finite sets	Normalized Hamming distance
Hyperlinearity	Finite-dimensional unitary matrices	Normalized Hilbert–Schmidt norm
MF approximation	Finite-dimensional unitary matrices	Operator norm

The term MF group also has a more restrictive usage in the operator-algebra literature. Schafhauser requires approximations that additionally reproduce the canonical trace and the norms of finite linear combinations in the left regular representation. He explicitly distinguishes this a priori stronger definition from the group-embedding definition above. His condition is equivalent to the canonical trace on the reduced group C*-algebra being an MF trace.

Using this stronger condition, Schafhauser proved that if two amenable groups have a common subgroup that is normal in both, their amalgamated free product is MF.

Open question

No non-MF group is known to exist, and it is an open question whether every group is MF.

Bachner, Dogon and Lubotzky describe this as a longstanding open problem in their 2026 account. They construct groups that cannot be approximated in the Schatten $1$-norm, while leaving the operator-norm question open.

One approach studies stability together with residual finiteness. A finitely generated group that is both MF and stable with respect to the operator norm must be residually finite. Consequently, a finitely generated group that is operator-norm stable but not residually finite would provide a non-MF group.

References

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• Properties of groups
• Operator algebras