Hyperbolic Groups and Non-Compact Real Algebraic Curves

TL;DR

This paper explores the structure and classification of non-compact real algebraic curves, using Fuchsian groups for uniformization and describing their moduli spaces as quotients of vector spaces by discrete groups.

Contribution

It provides a detailed description of the uniformization and moduli spaces of non-compact real algebraic curves, including their connected components and dimensions.

Findings

Connected components are homeomorphic to quotients of finite-dimensional real vector spaces by discrete groups.

Dimensions of the vector spaces are explicitly determined.

Uniformization of these curves is achieved via Fuchsian groups.

Abstract

In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs $(P,\tau)$, where $P$ is a compact Riemann surface with a finite number of holes and punctures and $\tau:P\to P$ is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.