Résume  Bernoulli percolation consists in erasing independently each edge of a graph G with some probability 1p and studying the connected components (called clusters) of this random graph. Of interest is the parameter p_c above which infinite clusters exist. We prove that the set of possible values for the percolation threshold p_c of Cayley graphs has a gap at 1 in the sense that there exists ε>0 such that for every Cayley graph G one either has p_c(G)=1 or p_c(G)≤1−ε. The proof builds on the new approach of DuminilCopin, Goswami, Raoufi, Severo & Yadin to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov's theorem on the structure of groups of polynomial growth of Breuillard, Green & Tao.
