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

We show that if a non-amenable, quasi-transitive, unimodular graph G has all degrees even then it has a factor-of-iid balanced orientation, meaning each vertex has equal in-and outdegree. This result involves extending earlier spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of quasi-transitive, unimodular graphs.As a consequence, we also obtain that when G is regular (of either odd or even degree) and bipartite, it has a factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov beyond transitive graphs.& COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).