No percolation at criticality on certain groups of intermediate growth

We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form $p_n(v,v)\le \exp [-\mathrm{\Omega }(n^{\gamma })]$ for some $\gamma >1\mathrm{/}2$. The result is new in the case that the graph is of intermediate volume growth.