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Let $G$ be a finite $p$-group. We prove that whenever the commutingprobability of $G$ is greater than $(2p^2 + p - 2)/p^5$, the unramified Brauergroup of the field of $G$-invariant functions is trivial. Equivalently, allrelations between commutators in $G$ are consequences of some universal ones.The bound is best possible, and gives a global lower bound of $1/4$ for allfinite groups. The result is attained by describing the structure of groupswhose Bogomolov multipliers are nontrivial, and Bogomolov multipliers of all oftheir proper subgroups and quotients are trivial. Applications include aclassification of $p$-groups of minimal order that have nontrivial Bogomolovmultipliers and are of nilpotency class $2$, a nonprobabilistic criterion forthe vanishing of the Bogomolov multiplier, and establishing a sequence ofBogomolov's absolute $\gamma$-minimal factors which are $2$-groups ofarbitrarily large nilpotency class, thus providing counterexamples to some ofBogomolov's claims. In relation to this, we fill a gap in the proof oftriviality of Bogomolov multipliers of finite simple groups.

Universal commutator relations, Bogomolov multipliers, and commuting probability