Gauglhofer, ThomasSemmler, Klaus-Dieter2010-12-032010-12-032010-12-03200510.1090/S1088-4173-05-00106-2https://infoscience.epfl.ch/handle/20.500.14299/61870Summary: We explicitly give $calT$, 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 $calT$ which is its moduli space. As a consequence, we see that on any hyperbolic Riemann surface, two closed geodesics of lengths smaller than $2operatornamearccosh(2)$ intersect at most once; two closed geodesics of lengths smaller than $2operatornamearccosh(3)$ are both non-dividing or intersect at most once.text::journal::journal article::research article