Let , define group generated by two operations

Assume are real linear independent, then is called a torus. In general we would orient the torus by assuming .

and are conformally equivalent if and only if and are conjugate in .

If and are conjugate then it is easy to prove the conclusion.

Suppose and . Since the universal cover of is , by the lifting lemmam, there exists a unique such that the following diagram commutes

From , . Then , which means .

Suppose , then from , . So there exists such that . Actually, the choice of does not depend on . This is because the group and act discontinouly on , then there is a neighborhood of such that remain the same.

So fix , is an open set. is also closed, since are compatible with the topology of and are continuous. So , which means for a fixed , such that . Thus . For the same sake, we have . and are conjugate in .

Consider . Suppose is generated by the mobius transformation and , is generated by and . Let us denote this four mobius transformations as and , .

Since , , . Then such that

Then . Since and are symmetric, must have inverse in . Thus . Since we require and be positive, then .

and are conformally equivalent if and only if in . The module space of torus is .

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