Fundamental group of uniquely ergodic Cantor minimal systems

We introduce the fundamental group $\mathcal{F}({\mathcal{R}}_{G,\varphi })$ of a uniquely ergodic Cantor minimal $G$-system ${\mathcal{R}}_{G,\varphi }$ where $G$ is a countable discrete group. We compute fundamental groups of several uniquely ergodic Cantor minimal $G$-systems. We show that if ${\mathcal{R}}_{G,\varphi }$ arises from a free action $\varphi$ of a finitely generated abelian group, then there exists a unital countable subring $R$ of $\mathbb{R}$ such that $\mathcal{F}({\mathcal{R}}_{G,\varphi })=R_{+}^{\times }$. We also consider the relation between fundamental groups of uniquely ergodic Cantor minimal ${\mathbb{Z}}^n$-systems and fundamental groups of crossed product $C^{\ast }$-algebras $C(X){⋊}_{\varphi }{\mathbb{Z}}^n$.