We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations of the surface with extra decorations. This covering graph is a skeleton for a space of suitably framed quadratic differentials on the surface, which in turn models the space of Bridgeland stability conditions for the 3-Calabi–Yau category associated to the marked surface. By showing that the relations in the covering groupoid are homotopically trivial when interpreted as loops in the space of stability conditions, we show that this space is simply connected.

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The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of RapoportZink spaces and of affine DeligneLusztig varieties. We prove a HodgeNewton decomposition for affine DeligneLusztig varieties and for the special fibers of RapoportZink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (HodgeNewton decomposability) which can be phrased in combinatorial terms. Second, we study the Shimura varieties in which every non-basic\sigma-isogeny class is HodgeNewton decomposable. We show that (assuming the axioms of He and Rapoport in Manuscr. Math. 152 (34): 317343, 2017) this condition is equivalent to nice conditions on either the basic locus or on

The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.

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