A desingularization of real differentiable actions of finite groups

We provide abelianizations of differentiable actions of finite groups on smooth real manifolds. De Concini-Procesi wonderful models for (local) subspace arrangements and a careful analysis of linear actions on real vector spaces are at the core of our construction. In fact, we show that our abelianizations have stabilizers isomorphic to elementary abelian 2-groups, a setting for which we suggest the term digitalization. As our main examples, we discuss the resulting digitalizations of the permutation actions of the symmetric group on real linear and projective spaces.