  We introduce notions of absolutely non-free and perfectly non-free group
actions and use them to study the associated unitary representations. We show
that every weakly branch group acts absolutely non-freely on the boundary of
the associated rooted tree. Using this result and the symmetrized diagonal
actions we construct for every countable branch group infinitely many different
ergodic perfectly non-free actions, infinitely many II$_1$-factor
representations, and infinitely many continuous ergodic invariant random
subgroups.
