Fundamental group of log terminal T-varieties

In this article, we introduce an approach to study the fundamental group of a log terminal T-variety. As applications, we prove the simply connectedness of the spectrum of the Cox ring of a complex Fano variety, we compute the fundamental group of a rational log terminal T-varieties of complexity one, and we study the local fundamental group of log terminal T-singularities with a good torus action and trivial GIT decomposition.