Abstract
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.
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C. M. Натанзон, Инвариантные прямы фуксовых групп, YMH 27 (1972), bьш. 4(166), 145–160. Engl. transl.: S. Natanzon, Invariant lines of Fuchsian groups, Russian Math. Surveys 27 (1972), no. 4, 161–177.
C. M. Натанзон, Модули вешественных алгебраикескин кривых, YMH 30 (1975), bьш. 1(181), 251–252. [S. Natanzon, Moduli of real algebraic curves, Uspekhi Mat. Nauk 30 (1975), 251–252 (in Russian)].
C. M. Натанзон, Пространства модуле вешественных кривых, Труды MMO 37 (1978), 219–253. Engl. transl.: S. Natanzon, Moduli spaces of real curves, Trans. Mosc. Math. Soc. 1 (1980), 233–272.
C. M. Натанзон, Модули римановых поверхностей, вещественных алгебраиче ских кривых и их супераналоги, MЦHMO, M., 2003. Engl. transl.: S. M. Natanzon, Moduli of Riemann Surfaces, Real Algebraic Curves, and their Superanalogs, Translations of Mathematical Monographs, Vol. 225, American Mathematical Society, Providence, RI, 2004.
C. M. Натанзон, Комплексный анализ, Римановы поверхности и интегриру-етые систеты, MЦHMO, M., 2018. [S. Natanzon, Complex Analysis, Riemann Surfaces and Integrable Systems, MCNMO, Moscow, 2018 (in Russian)].
J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, New York, 1994.
O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. (1940). no. 22, 1–197. Engl. transl.: O. Teichmüller, Extremal quasiconformal mappings and quadratic differentials, in: Handbook of Teichmüller Theory, Vol. V, Europ. Math. Soc., Zürich, 2016, pp. 321–483.
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In memory of Ernest Borisovich Vinberg
Sergey Natanzon is supported in part by the grant RFBR-20-01-00579.
Sergey Natanzon is deceased on December 7, 2020.
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NATANZON, S., PRATOUSSEVITCH, A. HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES. Transformation Groups 26, 631–640 (2021). https://doi.org/10.1007/s00031-021-09644-1
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DOI: https://doi.org/10.1007/s00031-021-09644-1