The isoperimetric profile of homogeneous Riemannian manifolds

We compute, up to a multiplicative constant, the isoperimeric profile of (n on-compact) homogeneous Riemannian manifolds by constructing "explicit" exh austions which give estimates for the distribution of the volume. For those Riemannian manifolds only three very different isoperimetric profiles exis t and the isoperimetric profile governs the asymptotic of the heat kernel d ecay on the diagonal and vice-versa. By discretisation, the isoperimetric p rofiles of finitely generated discrete subgroups of Lie groups are also com puted.