Fundamental group of uniquely ergodic Cantor minimal systems

We introduce the fundamental group F(R-G,R-phi) of a uniquely ergodic Cantor minimal G-system R-G,R- (phi) where G is a countable discrete group. We compute fundamental groups of several uniquely ergodic Cantor minimal G-systems. We show that if R-G,R-phi arises from a free action phi of a finitely generated abelian group, then there exists a unital countable subring R of R such that T(R-G,R-phi) = R-+(x). We also consider the relation between fundamental groups of uniquely ergodic Cantor minimal Z(n)-systems and fundamental groups of crossed product C*-algebras C(X) x(phi) Z(n). (C) 2012 Elsevier Inc. All rights reserved.