  We prove that Bernoulli bond percolation on any nonamenable, Gromov
hyperbolic, quasi-transitive graph has a phase in which there are infinitely
many infinite clusters, verifying a well-known conjecture of Benjamini and
Schramm (1996) under the additional assumption of hyperbolicity. In other
words, we show that $p_c<p_u$ for any such graph. Our proof also yields that
the triangle condition $\nabla_{p_c}<\infty$ holds at criticality on any such
graph, which is known to imply that several critical exponents exist and take
their mean-field values. This gives the first family of examples of one-ended
groups all of whose Cayley graphs are proven to have mean-field critical
exponents for percolation.
