Bogomolov Multiplier

• Bogomolov multiplier is a group-theoretic invariant that refines the Schur multiplier by focusing on degree-two cohomology classes that vanish on all bicyclic (abelian) subgroups.
• It is computed via homological models and nonabelian exterior squares, providing concrete criteria for vanishing or nonvanishing in various finite groups and p-groups.
• The invariant extends to Lie algebras, tensor categories, and physics, thus bridging classical group cohomology with modern applications in birational geometry and representation theory.

Searching arXiv for recent and foundational papers on the Bogomolov multiplier to ground the article in published work. arxiv_search(query="Bogomolov multiplier finite groups Lie algebras", max_results=10) The Bogomolov multiplier is a group-theoretic invariant that refines the Schur multiplier by isolating those degree-two cohomology classes that vanish on all abelian, equivalently bicyclic, subgroups. For a finite group $G$, it is usually denoted $B_0(G)$, although some papers write $B(G)$, and its homological incarnation is often denoted $\widetilde{B}_0(G)$. Its importance comes from Bogomolov’s theorem identifying it with the unramified Brauer group, or degree-two unramified cohomology, of $\mathbb C(V)^G$ for a faithful finite-dimensional complex representation $V$; consequently, nontrivial $B_0(G)$ obstructs rationality and even retract rationality in Noether’s problem (Chen et al., 2013, Kang et al., 2013, Kang, 2012).

1. Definition and basic interpretations

For a finite group $G$, the Bogomolov multiplier is the subgroup of the Schur multiplier $H^2(G,\mathbb Q/\mathbb Z)$ consisting of classes that restrict trivially to every abelian subgroup. One standard form is

$$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$

where $B_0(G)$0 ranges over abelian subgroups. Bogomolov’s simplification allows one to replace “all abelian subgroups” by bicyclic subgroups, meaning cyclic groups and direct products of two cyclic groups (Kang et al., 2013, Chen et al., 2013).

A cocycle-level formulation is often more concrete. If $B_0(G)$1, then $B_0(G)$2 precisely when

$B_0(G)$3

Equivalently, the restriction of $B_0(G)$4 to every abelian subgroup is trivial. This commuting-pair criterion is central in categorical and physical applications because it isolates cohomology classes that are globally nontrivial but invisible on commuting sectors (Davydov, 2013).

The birational meaning is fundamental. If $B_0(G)$5 is a faithful finite-dimensional complex representation of $B_0(G)$6, then $B_0(G)$7 is canonically isomorphic to the unramified cohomology, equivalently the unramified Brauer group in degree two, of $B_0(G)$8. In particular, $B_0(G)$9 obstructs rationality, stable rationality, and retract rationality of invariant fields. A recurrent misconception is that $B(G)$0 should prove rationality; the literature only supports the weaker statement that vanishing removes this specific unramified Brauer obstruction (Chen et al., 2013, Kang et al., 2013, Kang, 2012).

2. Homological models and Hopf-type formulas

A decisive advance in the subject is the replacement of cohomological restriction problems by explicit commutator calculus. Let

$B(G)$1

where $B(G)$2 is the nonabelian exterior square. Define

$B(G)$3

Then Moravec’s description identifies the homological Bogomolov multiplier as

$B(G)$4

and for finite groups one has

$B(G)$5

Thus vanishing of $B(G)$6 is equivalent to the statement that every element of the Schur multiplier is generated by wedges coming from commuting pairs (Donadze et al., 2015, Hatui, 2021).

Several equivalent commutator models are used in the literature. One prominent device is the group $B(G)$7, generated by $B(G)$8 and an isomorphic copy $B(G)$9, with $\widetilde{B}_0(G)$0 isomorphic to $\widetilde{B}_0(G)$1. Under this identification,

$\widetilde{B}_0(G)$2

which converts $\widetilde{B}_0(G)$3 into an explicit quotient inside a nilpotent commutator group (Chen et al., 2013).

For a free presentation $\widetilde{B}_0(G)$4, the Bogomolov analogue of the Hopf formula is

$\widetilde{B}_0(G)$5

where $\widetilde{B}_0(G)$6. A cohomological dual form is

$\widetilde{B}_0(G)$7

This formula makes precise the slogan that $\widetilde{B}_0(G)$8 measures commutator relations not forced by universal commutator identities together with triviality on commuting pairs (Hatui, 2021, Fernández-Alcober et al., 2016, Moravec, 2018).

The same viewpoint explains the “curly exterior square” $\widetilde{B}_0(G)$9, defined by quotienting $\mathbb C(V)^G$0 by the subgroup generated by wedges of commuting pairs. Then $\mathbb C(V)^G$1 is the kernel of the induced commutator map

$\mathbb C(V)^G$2

This formulation is especially effective in the theory of $\mathbb C(V)^G$3-groups of maximal class and in the study of universal commutator relations (Fernández-Alcober et al., 2016, Jezernik et al., 2013).

3. Structural theorems and permanence properties

The Bogomolov multiplier has strong functorial behavior under several natural constructions. For direct products,

$\mathbb C(V)^G$4

via restriction. The proof uses the decomposition of the Schur multiplier of a direct product and the fact that the mixed tensor term is generated by commuting pairs, hence disappears modulo $\mathbb C(V)^G$5 (Kang, 2012).

For coprime semidirect products $\mathbb C(V)^G$6 with $\mathbb C(V)^G$7, restriction induces

$\mathbb C(V)^G$8

Here $\mathbb C(V)^G$9 denotes the fixed subgroup for the conjugation action of $V$0 on $V$1, hence on $V$2. This result situates $V$3 within a coprime cohomological decomposition parallel to the corresponding statement for full group cohomology (Kang, 2012).

The exact-sequence formalism has also been sharpened. If $V$4 and $V$5, one has the exact sequence

$V$6

$V$7

This yields concrete criteria for inflation. In particular, $V$8 is surjective if and only if

$V$9

and it is the zero map if and only if

$B_0(G)$0

These formulas place the behavior of $B_0(G)$1 in extensions on the same footing as classical Schur-multiplier calculations (Hatui, 2021).

A separate line of work connects vanishing of $B_0(G)$2 with rigidity. For finite groups, $B_0(G)$3-rigidity means $B_0(G)$4, equivalently the absence of outer class-preserving automorphisms. Many rigid families have trivial Bogomolov multiplier, including symmetric groups, finite simple groups, $B_0(G)$5-groups of order at most $B_0(G)$6, $B_0(G)$7-groups with cyclic maximal subgroup, $B_0(G)$8-groups with cyclic subgroup of index $B_0(G)$9, abelian-by-cyclic groups, Blackburn groups, extraspecial $G$0-groups, and almost extraspecial $G$1-groups (Kang et al., 2013). The same paper also stresses a limitation: rigidity is not necessary for vanishing, since Burnside’s classical counterexamples to $G$2-rigidity of order $G$3 still satisfy $G$4 (Kang et al., 2013). This leaves open the general question whether every $G$5-rigid group has trivial Bogomolov multiplier.

Central extensions provide another permanence phenomenon. If $G$6 is metacyclic, or if $G$7 or $G$8 and $G$9, then every central extension

$H^2(G,\mathbb Q/\mathbb Z)$0

satisfies $H^2(G,\mathbb Q/\mathbb Z)$1. Since most finite simple groups have Schur multiplier of order at most $H^2(G,\mathbb Q/\mathbb Z)$2, this yields triviality of $H^2(G,\mathbb Q/\mathbb Z)$3 for central extensions of most finite simple groups (Donadze et al., 2015).

4. Finite $H^2(G,\mathbb Q/\mathbb Z)$4-groups: classifications, criteria, and extremal behavior

The most detailed computations occur for finite $H^2(G,\mathbb Q/\mathbb Z)$5-groups. Historically, groups of order at most $H^2(G,\mathbb Q/\mathbb Z)$6 have trivial Bogomolov multiplier, whereas order $H^2(G,\mathbb Q/\mathbb Z)$7 already exhibits exceptional families. For odd $H^2(G,\mathbb Q/\mathbb Z)$8, nontrivial $H^2(G,\mathbb Q/\mathbb Z)$9 occurs exactly in the isoclinism family $$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$0 among groups of order $$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$1 (Chen et al., 2013).

For groups of order $$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$2, $$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$3 odd, a nearly complete classification was first obtained using James’ isoclinism families. The 2013 result showed that for $$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$4,

$$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$5

and

$$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$6

with $$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$7 left unresolved but conjectured to have trivial multiplier (Chen et al., 2013). A later paper completed this investigation: for every nonabelian group $$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$8 of order $$B_0(G)=\ker \Bigl[H^2(G,\mathbb{Q}/\mathbb{Z})\to \prod_{A\subset G} H^2(A,\mathbb{Q}/\mathbb{Z})\Bigr],$$9, with $B_0(G)$00 odd,

$B_0(G)$01

Thus the unresolved families $B_0(G)$02 do in fact have trivial Bogomolov multiplier (Hatui, 2021).

For unitriangular groups, the situation is strikingly uniform. If

$B_0(G)$03

with $B_0(G)$04, then

$B_0(G)$05

The same vanishing holds for central products of unitriangular groups. This gives a positive answer to the Kang–Kunyavskiĭ problem for unitriangular groups and removes the standard unramified Brauer obstruction for these families (Michailov, 2013).

For $B_0(G)$06-groups of maximal class, there is an exact criterion. If $B_0(G)$07 has order $B_0(G)$08, lower central series $B_0(G)$09, and

$B_0(G)$10

then

$B_0(G)$11

If $B_0(G)$12 has positive degree of commutativity, this simplifies to

$B_0(G)$13

When $B_0(G)$14 is metabelian and the degree of commutativity is positive, the multiplier reduces to the commutator structure of $B_0(G)$15: $B_0(G)$16 where $B_0(G)$17 is a uniformizing element. This leads to explicit infinite families of maximal-class groups with nontrivial, and even large, Bogomolov multipliers (Fernández-Alcober et al., 2016).

The paper on universal commutator relations introduced $B_0(G)$18-minimal groups: finite groups with $B_0(G)$19 such that all proper subgroups and proper quotients have trivial Bogomolov multiplier. Such groups are $B_0(G)$20-groups, their Bogomolov multiplier has prime exponent, their Frattini subgroup is abelian, and they have Frattini rank at most $B_0(G)$21, or at most $B_0(G)$22 in nilpotency class at least $B_0(G)$23 (Jezernik et al., 2013). In nilpotency class $B_0(G)$24, there are exactly two $B_0(G)$25-minimal isoclinism families, represented by groups $B_0(G)$26 and $B_0(G)$27 of order $B_0(G)$28, with

$B_0(G)$29

These groups are the minimal class-$B_0(G)$30 examples of nontrivial Bogomolov multiplier (Jezernik et al., 2013).

The same paper proves the commuting-probability criterion

$B_0(G)$31

for finite $B_0(G)$32-groups, with sharp bound, and deduces the global statement

$B_0(G)$33

for arbitrary finite groups (Jezernik et al., 2013). This links a coarse probabilistic invariant to vanishing of the unramified Brauer obstruction.

There are also exponent bounds. If $B_0(G)$34 is metabelian, or $B_0(G)$35, or $B_0(G)$36 is nilpotent of class at most $B_0(G)$37, or $B_0(G)$38 is a $B_0(G)$39-Engel group, then

$B_0(G)$40

More precisely, the paper proves the stronger divisibility

$B_0(G)$41

in these four cases (Moravec, 2018).

5. Lie algebras, Lie rings, multiplicative Lie algebras, and unoriented variants

The Bogomolov multiplier has been extended beyond finite groups to several algebraic settings. For Lie algebras $B_0(G)$42, one defines the exterior square $B_0(G)$43, the canonical map

$B_0(G)$44

its kernel $B_0(G)$45, and the subalgebra

$B_0(G)$46

The Lie-algebraic Bogomolov multiplier is then

$B_0(G)$47

For finite-dimensional Lie algebras over a field $B_0(G)$48, this admits a cohomological realization

$B_0(G)$49

and one has $B_0(G)$50. A Hopf-type formula is available: $B_0(G)$51 for a free presentation $B_0(G)$52. This theory proves invariance under isoclinism and exhibits nilpotent class-$B_0(G)$53 Lie algebras with arbitrarily large Bogomolov multiplier dimension (Rai, 2023).

An earlier Lie-algebraic paper introduced the commutativity-preserving exterior product $B_0(G)$54 and computed many examples explicitly. It established

$B_0(G)$55

proved the Hopf-type formula

$B_0(G)$56

and showed vanishing for abelian Lie algebras, Heisenberg Lie algebras, and several classical simple complex Lie algebras, while identifying nontrivial examples among nilpotent Lie algebras of dimensions $B_0(G)$57 and $B_0(G)$58 (Rostami et al., 2018).

For finite $B_0(G)$59-groups and finite $B_0(G)$60-Lie rings linked by the Lazard correspondence, the Bogomolov multipliers agree. If $B_0(G)$61 is a finite $B_0(G)$62-group of class at most $B_0(G)$63 and $B_0(G)$64 its Lazard correspondent, then the paper proves that CP covers correspond and

$B_0(G)$65

as abelian groups (Rostami et al., 2019). This places the invariant naturally within the bridge between nilpotent group theory and Lie theory.

A more recent generalization treats multiplicative Lie algebras $B_0(G)$66, where the underlying set carries both a group law and a multiplicative Lie operation. There the Schur multiplier becomes

$B_0(G)$67

the Bogomolov multiplier is defined by restriction to abelian subalgebras, and its homological model is

$B_0(G)$68

A Hopf-type formula holds: $B_0(G)$69 and isoclinic multiplicative Lie algebras have isomorphic Bogomolov multipliers (Kumar et al., 2024).

There is also an unoriented topological analogue. The unoriented Schur multiplier is identified with

$B_0(G)$70

and the unoriented Bogomolov multiplier

$B_0(G)$71

is obtained by quotienting by classes represented by tori, Klein bottles, and projective-space pieces. It gives the complete obstruction to extending free actions of finite groups on closed unoriented surfaces to actions on $B_0(G)$72-manifolds, after excluding the subgroup generated by the trivial $B_0(G)$73-bundle over $B_0(G)$74 (Cruz et al., 2023).

6. Categorical, representation-theoretic, and physical avatars

The Bogomolov multiplier also appears in tensor-category theory. For a finite group $B_0(G)$75, let $B_0(G)$76 denote the Drinfeld centre. A cocycle $B_0(G)$77 defines a braided autoequivalence $B_0(G)$78, and $B_0(G)$79 is soft, meaning isomorphic to the identity as a $B_0(G)$80-linear functor, if and only if

$B_0(G)$81

This is exactly the Bogomolov condition. The paper proves an exact sequence

$B_0(G)$82

and identifies a natural subgroup

$B_0(G)$83

Thus $B_0(G)$84 is the cocycle part of the group of soft braided autoequivalences of the Drinfeld centre (Davydov, 2013).

A recent physics review emphasizes the same point from the perspective of projective representations. There, $B_0(G)$85 consists of cohomology classes represented by cocycles symmetric on commuting pairs but nontrivial in cohomology. Such classes characterize $B_0(G)$86-dimensional SPT phases that cannot be detected by string order parameters and, after gauging, produce distinct gapped phases with completely broken non-invertible $B_0(G)$87 symmetry. The paper constructs explicit lattice models and gives an example in which the ground-state degeneracy on a ring rises from $B_0(G)$88 without interfaces to $B_0(G)$89 with interfaces (Kobayashi et al., 16 Jul 2025).

Recent presentation-theoretic work extends computational access to broader families. For word labelled oriented graph groups, the homological Bogomolov multiplier $B_0(G)$90 is finitely generated with a computable generating set extracted from the underlying graph. This framework recovers vanishing for finitely presented Bestvina–Brady groups, even Artin groups, and right-angled Artin groups, while also computing Schur multipliers from the same presentations (Roy, 16 Apr 2025).

The modern picture is therefore bifocal. On one side, the Bogomolov multiplier is a birational obstruction encoded in unramified cohomology. On the other, it is a refined commutator invariant, computable by exterior-square methods, stable under isoclinism in several settings, and increasingly visible in tensor categories, representation theory, and quantum many-body physics. Open problems remain. The rigidity program leaves unresolved whether every $B_0(G)$91-rigid group has $B_0(G)$92, and permanence under central products is not settled in general (Kang et al., 2013). These questions suggest that the invariant is still best understood not as a closed chapter of the Schur multiplier, but as a precise measure of how global commutator and cohomological structure can remain invisible on all abelian localizations.