On non-uniqueness of percolation on nonamenable Cayley graphs

The aim of this Note is to prove a weak version of the conjecture of Benjamini and Schramm about phase of non-uniqueness for the Bernoulli bond percolation on nonamenable transitive graphs. We show that every nonamenable finitely generated group has a finite system of generators such that the Bernoulli bond percolation on the corresponding Cayley graph has a nonempty non-uniqueness phase. Together with previously known results, this gives a characterization of amenability of finitely generated groups in terms of uniqueness of percolation. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.