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HOME > Table of Contents and Abstracts > Vol. 60, No. 1
Tohoku Mathematical Journal
2008
March
SECOND SERIES VOL. 60, NO. 1
Tohoku Math. J.
60 (2008), 23-36
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Title
INTERPOLATION OF MARKOFF TRANSFORMATIONS ON THE FRICKE SURFACE
Author
Takeshi Sasaki and Masaaki Yoshida
(Received May 29, 2006, revised September 25, 2006) |
Abstract.
By the Fricke surfaces, we mean the cubic surfaces defined by the equation $p^2+q^2+r^2-pqr-k=0$ in the Euclidean 3-space with the coordinates $(p,q,r)$ parametrized by constant $k$. When $k=0$, it is naturally isomorphic to the moduli of once-punctured tori. It was Markoff who found the transformations, called Markoff transformations, acting on the Fricke surface. The transformation is typically given by $(p,q,r)\mapsto (r,q,rq-p)$ acting on $\boldsymbol{R}^3$ that keeps the surface invariant. In this paper we propose a way of interpolating the action of Markoff transformation. As a result, we show that one portion of the Fricke surface with $k=4$ admits a ${\rm GL}(2,\boldsymbol{R})$-action extending the Markoff transformations.
2000 Mathematics Subject Classification.
Primary 35J25; Secondary 28C15.
Key words and phrases.
Fricke surface, Markoff transformation.
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