  We consider the Bernoulli bond percolation process (with parameter $p$) on
infinite graphs and we give a general criterion for bounded degree graphs to
exhibit a non-trivial percolation threshold based either on a single
isoperimetric inequality if the graph has a bi-infinite geodesic, or two
isoperimetric inequalities if the graph has not a bi-infinite geodesic. This
new criterion extends previous criteria and brings together a large class of
amenable graphs (such as regular lattices) and non-amenable graphs (such
trees). We also study the finite connectivity in graphs satisfying the new
general criterion and show that graphs in this class with a bi-infinite
geodesic always have finite connectivity functions with exponential decay as
$p$ is sufficiently close to one. On the other hand, we show that there are
graphs in the same class with no bi-infinite geodesic for which the finite
connectivity decays sub-exponentially (down to polynomially) in the highly
supercritical phase even for $p$ arbitrarily close to one.
