Abstract
We study countable universes similar to a free action of a group G, where the similarity relation is the one recently defined by Poizat. It turns out that this is equivalent to the study of free semi-actions of G, with two universes being transformable iff one corresponding free semi-action can be obtained from the other by a finite alteration.
In the case of a free group G (in finitely many or countably many generators), a classification is given. The examples of non-commutative finitely generated free groups show that the most evident analogue of the Baldwin-Lachlan theorem is false for uncountably categorical universes.