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The Bogomolov multiplier of a group is the unramified Brauer group associatedto the quotient variety of a faithful representation of the group. This objectis an obstruction for the quotient variety to be stably rational. The purposeof this paper is to study these multipliers associated to nilpotent pro-$p$groups by transporting them to their associated Lie algebras. Special focus isset on the case of $p$-adic Lie groups of nilpotency class $2$, where weanalyse the moduli space. This is then applied to give information onasymptotic behaviour of multipliers of finite images of such groups of exponent$p$. We show that with fixed $n$ and increasing $p$, a positive proportion ofthese groups of order $p^n$ have trivial multipliers. On the other hand, weshow that by fixing $p$ and increasing $n$, log-generic groups of order $p^n$have non-trivial multipliers. Whence quotient varieties of faithfulrepresentations of log-generic $p$-groups are not stably rational. Applicationsin non-commutative Iwasawa theory are developed.

Irrationality of generic quotient varieties via Bogomolov multipliers