Parameters for Fuchsian Groups I: Signature (0, 4)

Abstract

This is the first of a series of notes presenting new parameters for certain torsion-free finitely generated Fuchsian and quasifuchsian groups. In this note we consider signature (0, 4). Other low signatures, as well as the general case, will be dealt with elsewhere. Every Fuchsian group of signature (0, 4), acting on the upper half-plane ∪, can be generated by four parabolic transformations, A,B,C,D, where the product ABCD = 1. Normalize so that AB has its attracting fixed point at ∞, its repelling fixed point at 0, and so that the fixed point of C is at 1. Let x be the fixed point of D, and let y be the fixed point of B. Then x > 1 and y < 0. We show that x and y serve as parameters for the deformation space of these groups (this is really two results, one having to do with Fuchsian, and the other with quasifuchsian groups). We also explicitly write the matrices A,B,C,D in PGL(2,ℝ) ⁺ (these are 2 × 2 real matrices with positive determinant) as functions of x and y; this gives an explicit example of a stratification (see [K-M]). We also construct an explicit fundamental domain for the Teichmüller modular group for signature (0, 4), and we identify the side pairing transformations.