Trace coordinates of Teichmüller space of Riemann surfaces of signature (0,4)

Abstract

We explicitly give T \mathcal {T} , the Teichmüller space of four-holed spheres (which we call X pieces) in trace coordinates, as well as its modular group and a fundamental domain for the action of this group on T \mathcal {T} which is its moduli space. As a consequence, we see that on any hyperbolic Riemann surface, two closed geodesics of lengths smaller than 2 arccosh ⁡ ( 2 ) 2\operatorname {arccosh}(2) intersect at most once; two closed geodesics of lengths smaller than 2 arccosh ⁡ ( 3 ) 2\operatorname {arccosh}(3) are both non-dividing or intersect at most once.