## Tohoku Mathematical Journal 2008 March SECOND SERIES VOL. 60, NO. 1

 Tohoku Math. J. 60 (2008), 23-36

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.