On a toroidalization for klt singularities

In this article, we prove a toroidalization principle for finite actions on klt singularities. As an application, we prove that the Jordan property for the regional fundamental group of klt singularities can be realized geometrically: by extracting a toric singularity over the klt germ. In the course of the proof, we will prove statements about finite actions on dual complexes and almost fixed points in the fibers of equivariant Fano type morphisms. Furthermore, we will prove that the rank of a fundamental group of the klt singularity is bounded above by its regularity.