On Veech groups of infinite superelliptic curves
Year: 2025
Type: article
Abstract: 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 C {\mathbb{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.
Source: Complex Manifolds
Author: Camilo Ramírez Maluendas
Institution: Universidad Nacional de Colombia
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $848
OpenAlex ID: W4411855194