Existentially Closed Measure-Preserving Actions of Approximately Treeable Groups

The paper proves that if Γ is approximately treeable then the model companion T*_Γ exists, via an open-mapping criterion (Theorem 5.26) and a careful construction using map–measure pairs and randomization along equivalence classes; see the statement and outline of Theorem 7.4 and its proof sketch in Section 7 (Theorem 7.4 and §7.2, §7.5) and the open-mapping framework in §5.5 . By contrast, the candidate solution assumes that approximate treeability (a group-level property defined via invariant measures on F(Γ), cf. Proposition 2.4 and Corollary 2.5) yields, for any given action, a large-measure subset on which the orbit relation is generated by an acyclic graphing and can be lifted to a free F_d-action; this step is not provided by the paper’s definition of approximate treeability and is unjustified . The subsequent pullback argument and the claim that the e.c. class is axiomatized without invoking the paper’s open-mapping criterion are also unsupported. Hence the paper’s result stands, but the model’s proof is flawed.