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abstract: '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.'
author:
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    Micha[ł]{} Gaczkowski$^1$, Przemys[ł]{}aw Górka$^1$[^1] and Daniel J. Pons$^2$[^2]\
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title: '**Symmetry and compact embeddings for critical exponents in metric-measure spaces**'
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[**Keywords**]{}: Sobolev spaces, metric-measure spaces, compact embedding

*Mathematics Subject Classification (2010):* 46E35; 30L99.

Introduction
============

Arising in the Calculus of Variations and PDE’s, the study of Sobolev spaces in Euclidean domains, and the embeddings between them, has been an active area of research for more than a century (see [@Adams] for the classical results, and [@naumann] for an overview on history). In the last fifty years, motivated by problems in Geometric Analysis, Physics and Topology, those studies have been generalized to functions on manifolds, with the extension to sections of vector bundles over those spaces, see [@aubin; @Hawking; @hebey; @lawson; @palais]. More recently, the study of metric-measure spaces[^3] 