  We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs,
and more generally graphs whose automorphism group has a nonunimodular
quasi-transitive subgroup. We prove that percolation on any such graph has a
non-empty phase in which there are infinite light clusters, which implies the
existence of a non-empty phase in which there are infinitely many infinite
clusters. That is, we show that $p_c<p_h \leq p_u$ for any such graph. This
answers a question of Haggstrom, Peres, and Schonmann (1999), and verifies the
nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996).
We also prove that the triangle condition holds at criticality on any such
graph, which implies that various critical exponents exist and take their
mean-field values.
  All our results apply, for example, to the product $T_k\times \mathbb{Z}^d$
of a $k$-regular tree with $\mathbb{Z}^d$ for $k\geq 3$ and $d \geq 1$, for
which these results were previously known only for large $k$. Furthermore, our
methods also enable us to establish the basic topological features of the phase
diagram for anisotropic percolation on such products, in which tree edges and
$\mathbb{Z}^d$ edges are given different retention probabilities. These
features had only previously been established for $d=1$, $k$ large.
