| In this talk, we study the infinite superelliptic curves as translation surfaces, which are branched covering of the complex plane with branching over infinitely many points. We give a characterization of the Veech group of such surfaces in terms of the matrices in \({\rm GL}_{+}(2,\mathbb{R})\), arising from the differential of the affine diffeomorphisms from \(\mathbb{C}\) to itself, permuting the branched points. We obtain necessary and sufficient conditions to guarantee that the Veech group of an infinite superelliptic curve is uncountable. We establish a trichotomy on the holonomy vector set and have a precise description of some countable groups that can appear as Veech group of an infinite superelliptic curve through the study of this trichotomy. We also construct and study several examples of interesting infinite superelliptic curves illustrating our results. |
