In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs (P, τ), where P is a compact Riemann surface with a finite number of holes and punctures and τ: P → 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.

中文翻译：

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双曲群和非紧实代数曲线

在本文中，我们研究非紧实代数曲线的空间，即对（P，τ），其中P是具有有限数量的孔和刺孔的紧Riemann曲面，并且τ：P → P是反全同性对合。我们描述了通过Fuchsian群对非紧实代数曲线的均一化。我们构造了非紧实代数曲线的空间并描述了它们的连通分量。我们证明了任何连接的分量对于离散维群的有限维实向量空间的商都是同胚的，并确定了这些向量空间的维数。