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, branching over infinitely many points. We provide a criterion for isomorphism between a special family of infinite superelliptic curves. We describe the geometry of saddle connections and holonomy vectors on these infinite superelliptic curves. In addition, we prove that the Veech group of an infinite superelliptic curve consists of matrices arising from the differentials of the affine mappings from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">C</m:mi> </m:math> {C} to itself, which permutes the branched points. We obtain necessary and sufficient conditions to guarantee that the Veech group of an infinite superelliptic curve is uncountable. Furthermore, we establish a trichotomy on the holonomy vector set and precisely characterize certain countable groups that can appear as Veech groups of an infinite superelliptic curve. Finally, we also construct and study several examples of interesting infinite superelliptic curves illustrating our results.