On essential freeness of actions of mixed identity-free groups

It was asked in [ACTT23] if every faithful, discrete, ergodic pmp action of a mixed identity-free (MIF) group is essentially free.

In this short note we give some positive evidence by showing that for a MIF group, any faithful generalized Bernoulli action and any affine action on a compact abelian group is essentially free.

This gives some classes of discrete ergodic actions with this property beyond the totally ergodic or compact case.