Rational solutions of E~=l aix~ = dXIX2Xa and simple closed geodesics on Fricke surfaces
Mark Sheingorn · 1988
equations, a (1.0) Laix~ = dXIX2Xa; (al,a2,aa,d) = 1. i=l We find that rational solutions of any such equation lie in disjoint, essen tially binary trees. (Actually the same trees exist when the ai and d and the solutions are taken to be real. It is the number-theoretic bent of this paper, and the observation that the rational solutions of the (general) case (1.0) being the appropriate generalization of the integral solutions of the (special) case of Markov's equation that led us to stress the word rational in the above text.) For a given equation, each tree is associated with a single Riemann surface of signature [0; 2, 2, 2, 00] and the solutions of (1.0) in that tree characterize the simple closed geodesics of that surface. From the solutions, there is an explicit algorithm to construct a Fuchsian group q,o representing the surface and the hyperbolic matrices whose axes project to all simple closed geodesics on the surface. Also from our presentation of q,o, we obtain pre sentations of 8 0 and wo; these Fuchsian subgroups of q,o represent surfaces of type [1; 00] and [0; 00, 00, 00, 00] respectively. The same hyperbolic axes project to all simple closed geodesics on these surfaces also. (Surfaces with these signatures are called Fricke surfaces.) Finally, we give an explicit con struction of all the rational trees of (1.0), concluding with some remarks about integral solutions.