On differentiable compactifications of the hyperbolic plane and algebraic actions of $${rm SL}_2(mathbb{R})$$ on surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2;R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. The online version of the original article can be found under doi: .