  A celebrated theorem of Kanai states that quasi-isometries preserve
isoperimetric inequalities between uniform Riemannian manifolds (with positive
injectivity radius) and graphs. Our main result states that we can study the
(Cheeger) isoperimetric inequality in a Riemann surface by using a graph
related to it, even if the surface has injectivity radius zero (this graph is
inspired in Kanai's graph, but it is different from it). We also present an
application relating Gromov boundary and isoperimetric inequality.
