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We give a complete answer to which homogeneous groups admit homogeneousdistances for which the Besicovitch Covering Property (BCP) holds. Inparticular, we prove that a stratified group admits homogeneous distances forwhich BCP holds if and only if the group has step 1 or 2. These results areobtained as consequences of a more general study of homogeneous quasi-distanceson graded groups. Namely, we prove that a positively graded group admitscontinuous homogeneous quasi-distances satisfying BCP if and only if any twodifferent layers of the associated positive grading of its Lie algebra commute.The validity of BCP has several consequences. Its connections with the theoryof differentiation of measures is one of the main motivations of the presentpaper. As a consequence of our results, we get for instance that a stratifiedgroup can be equipped with some homogeneous distance so that thedifferentiation theorem holds for each locally finite Borel measure if and onlyif the group has step 1 or 2. The techniques developed in this paper allow alsous to prove that sub-Riemannian distances on stratified groups of step 2 orhigher never satisfy BCP. Using blow-up techniques this is shown to imply thaton a sub-Riemannian manifold the differentiation theorem does not hold for somelocally finite Borel measure.

Besicovitch Covering Property on graded groups and applications to measure differentiation