On Correspondences between Once Punctured Tori and Closed Tori: Fricke Groups and Real Lattices
Ryuji Abe · Tokyo Journal of Mathematics · 2000
We consider the Teichmuller space of the closed torus and the Teichm\"uller space of the once punctured torus.It is well-known that the former can be identified with the upper half- plane and that several coordinate systems can be introduced to the latter.This is the first part of a series of papers in which we investigate explicit relations between these two Teichmuller spaces.In this paper based on a correspondence of subsets of these spaces we will give an explicit construction of a holomorphic mapping between a once punctured torus and a closed torus. We use throughout the convention that an elementWe consider a Fuchsian group $G$ consisting of M\"obius transformations of $PSL(2, R)$ and having the following properties:In the definition above $\Gamma=\langle A,$ $B$ ) is the free group generated by $A,$ $B$ and tr denotes the trace of a matrix.We consider a once punctured torus which is uniformized by a Fricke group $\Gamma$ and take a normalized form for the presentation of $\Gamma$ (see \S 5).By using the quantities $X=trA,$ $Y=trB$ and $Z=tr$ AB, the above description of the Fricke group is characterized by $X^{2}+Y^{2}+Z^{2}=XYZ$ and $X,$ $Y,$ $Z>2$ .Moreover, we obtain the following theorem (see [W]).THEOREM 1.1 (Fricke [F], Keen [K]).The Teichmuller space $\mathcal{T}_{1,1}$ of the once punc- tured torus is the sublocus of $X^{2}+Y^{2}+Z^{2}=XYZ$ with $X,$ $Y,$ $Z>2$.