Given a nonamenable transitive graph, does the plus state of the Ising model at low temperature stochastically dominate a high-density Bernoulli percolation? We'll discuss this question (asked by Liggett and Steif) and other questions of stochastic domination, and how amenability of lack thereof plays a role. Time permitting we'll also discuss invariant monotone coupling and applications to finitary codings. Joint work with Gourab Ray.