Strict monotonicity of percolation thresholds under covering maps

We answer a question of Benjamini and Schramm by proving that under reasonable conditions, quotienting a graph strictly increases the value of its percolation critical parameter $p_c$. More precisely, let $\mathcal{G}=(V,E)$ be a quasi-transitive graph with $p_c(\mathcal{G})<1$, and let $G$ be a nontrivial group that acts freely on $V$ by graph automorphisms. Assume that $\mathcal{H}:=\mathcal{G}/G$ is quasi-transitive. Then one has $p_c(\mathcal{G})<p_c(\mathcal{H})$. We provide results beyond this setting: we treat the case of general covering maps and provide a similar result for the uniqueness parameter $p_u$, under an additional assumption of boundedness of the fibres. The proof makes use of a coupling built by lifting the exploration of the cluster, and an exploratory counterpart of Aizenman-Grimmett's essential enhancements.