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