Measure equivalence embeddings of free groups and free group factors

The paper proves the group-level ME-embedding (Theorem A) and its weak mixing and stable strong ergodicity via an explicit skew-product and two key propositions: a fundamental-domain criterion relying on ‖λ(ν)‖ < 1/3 (Proposition 3.7) and a “random strong ergodicity” result (Proposition 3.6), combined with the correspondence transfer principle (Proposition 2.4). It also proves the II1-factor characterization (Theorem B) using a concrete cocycle construction and Connes’ averaging to get a norm < 1/3 (Theorem 5.6 + Proposition 5.7). By contrast, the candidate solution assumes, without proof, a crucial compression estimate ∥πY(ν)∥ ≤ 3‖λG(ν)‖ and a general tensor-norm inequality to deduce strong (and stable) ergodicity—claims that are neither established nor used in the paper. It also does not justify the dissipativity/fundamental-domain for the F2-action, which in the paper hinges on the (2k−1)−1 threshold. Hence the conclusions match the paper but the model’s proof contains unproven steps and gaps.