Interpolation of Markoff transformations on the Fricke surface

摘要

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) -> (r, q, rq - p) acting on 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 GL(2, R)-action extending the Markoff transformations.