CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 9, Pages 46–75 (April 26, 2005) S 1088-4173(05)00106-2 TRACE COORDINATES OF TEICHM ¨ ULLER SPACE OF RIEMANN SURFACES OF SIGNATURE (0, 4) THOMAS GAUGLHOFER AND KLAUS-DIETER SEMMLER Abstract. We explicitly give T , the Teichm¨ uller 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 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) intersect at most once; two closed geodesics of lengths smaller than 2 arccosh(3) are both non-dividing or intersect at most once. 1. Introduction This paper deals with Teichm¨ uller space and the Riemann moduli problem for surfaces of signature (0, 4) which we call X pieces. The Riemann moduli problem is to describe the space of isomorphism classes of Riemann surfaces of a given signature which is known as the moduli space. Teichm¨ uller space can be defined in many different ways, for instance as the space of isotopy classes of marked complex structures of the Riemann surface, where two structures define the same point in Teichm¨ uller space if there exists a holomorphic homeomorphism homotopic to the identity leading from one to the other. Once Teichm¨ uller space is well known, we can construct the modular group or mapping class group and build the quotient of Teichm¨ uller space with it to get the moduli space. In this paper we take the view that a Riemann surface is given by a Fuchsian group, namely a discrete subgroup of SL(2, R) acting on the upper half plane H by M¨ obius transformations and by which we quotient H to obtain the structure, which is determined by this subgroup up to conjugation. We choose SL(2, R) rather than PSL(2, R), as is usually done in literature, because this gives us additional information on direction of geodesics (see section 2). However, as we choose the traces of the generating elements to be positive, these two approaches are equivalent (see [SS92]). In this language we can define Teichm¨ uller space as the space of endomorphisms of the Fuchsian subgroup into SL(2, R) up to conjugation that preserve parabolic elements and whose image is discrete. As the referee pointed out, this approach is quite the same as the one taken in [Kee77] (based upon previous work; see [Kee65, Kee66, Kee71, Kee73]) where Keen gives rough fundamental domains for the action of modular groups on Teichm¨ uller spaces for various signatures including (0, 4). However, our way to show that there Received by the editors September 3, 2003 and, in revised form, February 8, 2005. 2000 Mathematics Subject Classification. Primary 32G15, 30F35; Secondary 11F06. The authors were supported in part by the Swiss National Science Foundation, SNSF Grant #2100-065270, Teichm¨ uller Spaces in Trace coordinates and Modular groups . c 2005 American Mathematical Society Reverts to public domain 28 years from publication 46