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Mat. Zametki, 1976, Volume 19, Issue 3, Pages 419–428 (Mi mz7760)  

A group of transformations connected with the Markov cubic surface

V. V. Ermakov

M. V. Lomonosov Moscow State University

Abstract: 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.

Full text: PDF file (664 kB)

English version:
Mathematical Notes, 1976, 19:3, 256–261

Bibliographic databases:

UDC: 513
Received: 02.07.1975

Citation: V. V. Ermakov, “A group of transformations connected with the Markov cubic surface”, Mat. Zametki, 19:3 (1976), 419–428; Math. Notes, 19:3 (1976), 256–261

Citation in format AMSBIB
\Bibitem{Erm76}
\by V.~V.~Ermakov
\paper A~group of transformations connected with the Markov cubic surface
\jour Mat. Zametki
\yr 1976
\vol 19
\issue 3
\pages 419--428
\mathnet{http://mi.mathnet.ru/mz7760}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=407034}
\zmath{https://zbmath.org/?q=an:0347.14008}
\transl
\jour Math. Notes
\yr 1976
\vol 19
\issue 3
\pages 256--261
\crossref{https://doi.org/10.1007/BF01437860}


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