We determine the set of connected components of minuscule affine Deligne-Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne-Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport-Zink spaces.2000 Mathematics Subject Classification. 20G25, 14G35.

We prove the Fargues-Rapoport conjecture for p-adic period domains: for a reductive group G over a p-adic field and a minuscule cocharacter µ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G, µ) is fully Hodge-Newton decomposable.Contents 24 7. Asymptotic geometry of the admissible locus 28 References 30

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We propose a new stratification of the reduced subschemes of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties that highlights the relation between the geometry of these spaces and the action of the associated automorphism group. We show that this provides a joint group-theoretic interpretation of wellknown stratifications which only exist for special cases such as the Bruhat-Tits stratification of Vollaard and Wedhorn, the semi-module stratification of de Jong and Oort, and the locus where the a-invariant is equal to 1.

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