Composantes connexes géométriques de la tour des espaces de modules de groupes $p$-divisibles

Let M̆ be an unramified Rapoport-Zink space of EL type or unitary/symplectic PEL type. Let (M̆K)K be the tower of Berkovich’s analytic spaces classifying the level structures over the generic fiber of M̆. In [Che13], we have defined a determinant morphism detK from the tower (M̆K)K to a tower of Berkovich’s analytic spaces of dimension 0 associated to the cocenter of the reductive group related to the space M̆. Suppose that the Newton polygon and Hodge polygon related to M̆ don’t touch each other except their end point. And suppose that a conjecture on the set of connected components of the reduced special fiber of M̆ holds. Then we prove that the geometric fibers of the determinant morphism detK are the geometrically connected components of M̆K . The conjecture in the hypothesis will be confirmed in a paper in preparation by Kisin, Viehmann and the author.