# Fuchsian group which is derived from a division quaternion algebra, Mixing flows on the quotient space

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:

1. Is there any idea or a proof in the general case somewhere?
2. 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 !

• A general proof that an arithmetic Fuchsian group derived from a quaternion division algebra is cocompact appears in Maclachlan and Reid's book (The Arithmetic of Hyperbolic 3-Manifolds). It is Theorem 8.12. In fact, this proof applies to discrete subgroups of products of SL(2,R) and SL(2,C) arising from quaternion algebras as well. Thus, for instance, their theorem implies your question for arithmetic Kleinian groups. – Ben Linowitz Apr 9 '17 at 13:42