Qualitative graph limits and the space of Cantor actions

Recently, I defined a notion of limit for finite Schreier graphs. It turns out that the natural limit objects are stable continuous group actions on the Cantor set. The limit notion leads to the definition of qualitative weak equivalence (this notion has already been defined by Matui and Li for minimal actions of the integers). I will show some results on the space of weak equivalence classes and will allude to some applications in the theory of local distributed algorithms.