Noncommutative marked surfaces II: tagged triangulations, clusters, and their symmetries

The aim of the paper is to define noncommutative cluster structure on several algebras $\mathcal{A}$ related to marked surfaces possibly with orbifold points of various orders, which includes noncommutative clusters, i.e., embeddings of a given group $G$ into the multiplicative monoid ${\mathcal{A}}^{\times }$ and an action of a certain braid-like group $B{r}_{\mathcal{A}}$ by automorphisms of each cluster group in a compatible way. For punctured surfaces we construct new symmetries, noncommutative tagged clusters and establish a noncommutative Laurent Phenomenon.