On Veech groups of infinite superelliptic curves

We study infinite superelliptic curves as translation surfaces and explore their Veech groups. These objects are branched covering of the complex plane with branching over infinitely many points. We provide a criterion for isomorphism between a special family of infinite superelliptic curves. We show geometric descriptions of saddle connections and holonomy vectors on these infinite superelliptic curves. We prove that the Veech group of an infinite superelliptic curve are all the matrices arising from the differential of the affine mappings $\mathbb{C}$ 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 from it, we give a precise characterization of some countable groups that can appear as Veech group of an infinite superelliptic curve. We also construct and study several examples of interesting infinite superelliptic curves illustrating our results.