Suppose a Fuchsian group $\Gamma$ is derived from a division quaternion algebra. Then the quotient space $\Gamma\backslash \mathcal{H}$ is compact.

I am reading the book "Fuchsian Groups" of Svetlana Katok. In Theorem 5.4.1 (as above), there is only a proof for the simplest case when $A$ is a division algebra over $\mathbb{Q}$: $ A=\left ( \frac{a,b}{\mathbb{Q}} \right ), a>0, \mathcal{O}=\left \{ x\in A | x_0, x_1, x_2 , x_{3}\in \mathbb{Z} \right \}$

**My question:**

- Is there any idea or a proof in the general case somewhere?
- Are there some special properties of mixing flows on that surface which is wrong on compact hyperbolic surfaces in general?

Hoping someone can help me. Thanks in advance !