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

Gauglhofer, Thomas; Semmler, Klaus-Dieter

doi:10.1090/S1088-4173-05-00106-2

Trace coordinates of Teichmüller space of Riemann surfaces of signature $(0,4)$
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by Thomas Gauglhofer and Klaus-Dieter Semmler

Conform. Geom. Dyn. 9 (2005), 46-75

DOI: https://doi.org/10.1090/S1088-4173-05-00106-2

Published electronically: April 26, 2005

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Abstract:

We explicitly give $\mathcal {T}$, 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 $\mathcal {T}$ which is its moduli space. As a consequence, we see that on any hyperbolic Riemann surface, two closed geodesics of lengths smaller than $2\operatorname {arccosh}(2)$ intersect at most once; two closed geodesics of lengths smaller than $2\operatorname {arccosh}(3)$ are both non-dividing or intersect at most once.

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