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In this paper we characterise the action of the absolute Galois group on thegeometric finite cyclic groups without \'etale factorization of stack inertiaof the profinite geometric fundamental group of moduli spaces of marked curves.As a complementary result, we give the same action on prime order profiniteelements in genus 2.

Composantes irr\'eductibles de lieux sp\'eciaux d'espaces de modules de courbes, action galoisienne en genre quelconque