THE UNIFORM GARDNER CONJECTURE AND ROUNDING BOREL FLOWS

The paper proves that a finitely generated amenable group has the uniform Gardner property iff it does not admit D∞ ≅ (Z/2Z)*(Z/2Z) as a quotient by a finite subgroup (Theorem 1.4), via: (i) a negative direction using the impossibility of measurable invariant end selections in totally ergodic D∞-actions (Lemma 4.1) and a construction of Γ-uniform sets that are Γ-equidecomposable but not measurably equidecomposable when Γ maps onto D∞ modulo a finite subgroup (Lemma 4.2 and Claim 4.3), and (ii) a positive direction for one-ended and finite-by-ℤ groups using bounded flows, rounding to integral flows on connected toasts (Theorem 3.1, Corollary 3.4), and Følner tilings to implement measurable equidecompositions (Proposition 4.5), finishing with the two-ended classification (Poénaru; Scott–Wall) in the proof of Theorem 1.4 . The candidate solution mirrors these steps closely: the D∞ obstruction via end selection, construction of Γ-uniform counterexamples, rounding flows on connected toasts, and the same two-ended reduction. Minor slips include stating gap sizes “2 or 3” instead of “1 or 2” in the maximal independent set on the line and asserting connected toasts for every ℤ-action (the paper instead uses a measurable end selection in the ℤ-quotient case), but these do not affect the main argument or conclusion.