  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
$\varepsilon_0>0$ such that for every Cayley graph $G$ one either has
$p_c(G)=1$ or $p_c(G) \leq 1-\varepsilon_0$. The proof builds on the new
approach of Duminil-Copin, 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.
