We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster. This allows us to deduce from earlier results that critical percolation on any graph in this class does not have any infinite clusters. The result is new when the graph in question is either amenable or nonunimodular.
A note about critical percolation on finite graphs
Critical percolation in the plane
Exponential growth of products of non-stationary Markov-dependent matrices
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