A group of transformations connected with the Markov cubic surface

V. V. Ermakov

Matematičeskie zametki, Tome 19 (1976) no. 3, pp. 419-428

Résumé

Let $V$ be the surface given by the equations \begin{gather*} x_1^2+x_2^2+x_3^2=3x_1x_2x_3; \\ x_1>0,x_2>0,x_3>0. \end{gather*} Let $V(R)$ and $V(Z)$ be its real and integral points respectively, and $G$ the group of transformations generated by $t_1$,$t_2$,$t_3$, where \begin{gather*} t_1(x_1,x_2,x_3)=(3x_2x_3-x_1,x_2,x_3) \\ t_2(x_1,x_2,x_3)=(x_1,3x_1x_3-x_2,x_3) \\ t_3(x_1,x_2,x_3)=(x_1,x_2,3x_1x_2-x_3) \end{gather*} It is shown in this paper that $G$ acts freely on $V(Z)$. On $V(R)$, $G$ acts discretely; we construct a fundamental domain, and describe the fixed points.