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We obtain a compact Sobolev embedding for $H$-invariant functions in compactmetric-measure spaces, where $H$ is a subgroup of the measure preservingbijections. In Riemannian manifolds, $H$ is a subgroup of the volume preservingdiffeomorphisms: a compact embedding for the critical exponents follows. Theresults can be viewed as an extension of Sobolev embeddings of functionsinvariant under isometries in compact manifolds.

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