Symmetry and compact embeddings for critical exponents in metric-measure spaces

We obtain a compact Sobolev embedding for H-invariant functions in compact metric-measure spaces, where H is a subgroup of the measure preserving bijections. In Riemannian manifolds, H is a subgroup of the volume preserving diffeomorphisms: a compact embedding for the critical exponents follows. The results can be viewed as an extension of Sobolev embeddings of functions invariant under isometries in compact manifolds.