Special Folding of Quivers and Cluster Algebras

We give a precise definition of folded quivers and folded cluster algebras. We give many examples of including some with finite mutation structure that do not have analogues in the unfolded cases. We relate these examples to the finite mutation type quivers $X_6$ and $X_7$. We also construct a folded cluster algebra associated to triangulations of punctured surface which allow for triangulations self-folded triangles. We give a simple construction of a folded cluster algebra for which the cluster complex is a generalized permutohedron.