Interpolation of Markoff Transformations on the Fricke Surface

By the Fricke surfaces, we mean the space defined by the equation p2 + q2 + r2 − pqr− k = 0 in the space R(p, q, r). 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) → (r, q, rq − p) acting on R3 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 the space {(p, q, r); p2+q2+r2−pqr−4 = 0, p > 2, q > 2, r > 2} admits a GL(2,R)-action extending the Markoff transformations.