WEAK AMENABILITY OF FREE PRODUCTS OF HYPERBOLIC AND AMENABLE GROUPS

The paper establishes that if G is amenable and H is hyperbolic, then G * H is weakly amenable, via orbit equivalence (OE) invariance of the Cowling–Haagerup constant, Ornstein–Weiss orbit equivalence to Z, and hyperbolicity of Z * H leading to weak amenability; see the statement and proof of Proposition 1.1 and Section 2 of the uploaded note . However, the proof as written implicitly uses G ∼OE Z for all amenable G without addressing the finite case (where G cannot be OE to Z via free ergodic actions), leaving a small gap. The model’s solution is essentially the same argument but explicitly splits off the finite case (where G * H is directly hyperbolic) and thus is complete.