On non-uniqueness of percolation on nonamenable Cayley graphs

The aim of this Note is to prove a weak version of the conjecture of Benjam ini and Schramm about phase of non-uniqueness for the Bernoulli bond percol ation on nonamenable transitive graphs. We show that every nonamenable fini tely generated group has a finite system of generators such that the Bernou lli bond percolation on the corresponding Cayley graph has a nonempty non-u niqueness phase. Together with previously known results, this gives a chara cterization of amenability of finitely generated groups in terms of uniquen ess of percolation. (C) 2000 Academie des sciences/Editions scientifiques e t medicales Elsevier SAS.