According to the wikipedia page on "Isoperimetric dimension", the isoperimetric dimension is invariant under quasi-isometries, even between manifolds and graphs:

"[...] the isoperimetric dimension is preserved by quasi isometries, both by quasi-isometries between manifolds, between graphs, and even by quasi isometries carrying manifolds to graphs, with the respective definitions."

Does someone know the precise definitions of this, and references on where to find this kind of result?