Critical percolation on any nonamenable group has no infinite clusters

We show that independent percolation on any Cayley graph of a nonamenable g roup has no infinite components at the critical parameter. This result was obtained by the present authors earlier as a corollary of a general study o f group-invariant percolation. The goal here is to present a simpler self-c ontained proof that easily extends to quasi-transitive graphs with a unimod ular automorphism group. The key tool is a "mass-transport" method, which i s a technique of averaging in nonamenable settings.