Fundamental group of uniquely ergodic Cantor minimal systems

We introduce the fundamental group F(RG,φ) of a uniquely ergodic Cantor minimal G-system RG,φ where G is a countable discrete group. We compute fundamental groups of several uniquely ergodic Cantor minimal G-systems. We show that if RG,φ arises from a free action φ of a finitely generated abelian group, then there exists a unital countable subring R of R such that . We also consider the relation between fundamental groups of uniquely ergodic Cantor minimal Zn-systems and fundamental groups of crossed product C⁎-algebras C(X)⋊φZn.