# Arithmetic Fuchsian group

Dear all,

I have the following questions: Are all Fuchsian groups of signature $(0;2,2,2,\infty)$ arithmetic? What is known about the trace fields of these groups?

Best, K.

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## 1 Answer

The answer is NO. Any four times punctured sphere admits two involutions, the quotient by which is an orbifold of the signature you describe. Similarly, you can take a punctured torus, and the quotient by the elliptic involution is one of your surfaces. Conversely, you can cover one of them by a torus or a four-times-punctured sphere. Since very few tori are arithmetic, same is true of your class of groups.

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