Factor-of-iid balanced orientation of non-amenable graphs

The paper proves that every non-amenable, quasi-transitive, unimodular graph with all degrees even admits a factor-of-iid balanced orientation via (i) a reduction to perfect matchings in an auxiliary bipartite construction G*, (ii) a spectral gap on the Bernoulli graphing for such graphs, (iii) spectral-to-expansion, and (iv) Lyons–Nazarov’s measurable matching on expanding graphings; the matching is then translated back to a balanced orientation. The candidate solution mirrors this pipeline essentially step-for-step: the G* reduction, Bernoulli graphing equivalence, spectral gap (Theorems 18–19 leading to Theorem 2), Cheeger-type expansion, transfer of expansion to the auxiliary graphing, and Lyons–Nazarov to get the matching, yielding the factor-of-iid balanced orientation. Minor presentation differences (e.g., orientation convention and measure normalization in the auxiliary graphing) do not affect correctness. See Theorem 1 statement and setup, the Bernoulli graphing equivalence, the spectral gap result and its use in the proof of Theorem 1, and the matching theorems for graphings .