Abstract
We study the action of the arithmetic Galois group on the geometric inertia subgroups of the fundamental group, in a tame but typically stacky context. The problem is analogous but more involved than the by now fairly well-understood situation of the procyclic inertia subgroups associated with the components of a divisor with normal crossings. A significant part of the text is devoted to introducing the necessary tools for a study which is in part motivated by and applied to the important example of the moduli stacks of curves, where the geometric inertia groups correspond to the automorphisms of algebraic curves of the type classified by the stack.
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