Critical site percolation dies on (almost) every quasi-transitive graph

Benjamini and Schramm conjectured in 1996 that critical site percolation has no infinite cluster on any quasi-transitive graph with p_c < 1. We prove this for quasi-transitive graphs with polynomial growth along a sequence of radii, for all quasi-transitive graphs of exponential growth, and for quasi-transitive graphs of subexponential growth whose simple random walk satisfies p_n(v, v) ≤ exp(−c n^γ) for some γ ∈ (1/2, 1]. The exponential case includes a complete proof of the site, quasi-transitive form of Timár's theorem for nonunimodular automorphism groups. The conjecture therefore remains open only for quasi-transitive graphs of intermediate growth without such a return-probability bound. On the graphs covered, the percolation probability is continuous on [0, 1]; and for site probabilities constant on the orbits of an automorphism group with finitely many orbits, the set of percolating parameters is open, except possibly for nonunimodular groups with at least two orbits on nonamenable graphs. This includes mixed site–bond percolation.