# INTERPOLATION OF MARKOFF TRANSFORMATIONS ON THE FRICKE SURFACE

## 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 \$\textrm{GL}(2,\boldsymbol{R})\$-action extending the Markoff transformations.

## Journal

• Tohoku Mathematical Journal, Second Series

Tohoku Mathematical Journal, Second Series 60(1), 23-36, 2008

Tohoku University

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