In this paper, we study percolation on finite Cayley graphs. A conjecture of Benjamini says that the critical percolation p_c of such a graph can be bounded away from one, for any Cayley graph satisfying a certain diameter condition. We prove Benjamini's conjecture for some special classes of groups. We also establish a reduction theorem, which allows us to build Cayley graphs for large groups without increasing p_c.
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