  In this paper we study percolation on a roughly transitive graph G with
polynomial growth and isoperimetric dimension larger than one. For these graphs
we are able to prove that p_c < 1, or in other words, that there exists a
percolation phase. The main results of the article work for both dependent and
independent percolation processes, since they are based on a quite robust
renormalization technique. When G is transitive, the fact that p_c < 1 was
already known before. But even in that case our proof yields some new results
and it is entirely probabilistic, not involving the use of Gromov's theorem on
groups of polynomial growth. We finish the paper giving some examples of
dependent percolation for which our results apply.
