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By the Fricke surfaces, we mean the space defined by the equation p2 + q2 + r2 ¡pqr ¡k = 0 in the space R3(p;q;r). When k = 0, it is naturally isomorphic to the moduli of once-punctured tori. It was Marko who found the transformations, called Marko transformations, acting on the Fricke surface. The transformation is typically given by (p;q;r) ! (r;q;rq ¡ p) acting on R3 that keeps the surface invariant. In this paper we propose a way of interpolating the action of Marko transformation. As a result, we show that the space f(p;q;r);p2+q2+r2¡pqr¡4 = 0;p > 2;q > 2;r > 2g admits a GL(2;R)-action extending the Marko transforma- tions.

Interpolation of Markoff transformations on the Fricke surface