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We prove that critical percolation has no infinite clusters almost surely onany unimodular quasi-transitive graph satisfying a return probability upperbound of the form $p_n(v,v) \leq \exp\left[-\Omega(n^\gamma)\right]$ for some$\gamma>1/2$. The result is new in the case that the graph is of intermediatevolume growth.

No percolation at criticality on certain groups of intermediate growth