I am looking to understand a citation about the connection of quaternion algebra over number fields which when embedded into $\mathbb{C}$, leads to a discrete subgroup of $SL\_2(\mathbb{C})$ which causes tilings of the hyperbolic 3-manifold $\mathbb{H}^3$. The author mentions the chapter 4 in the French book M.-F. Vigneras -"Arithmétique des algèbres de quaternions" for some computations of the fundamental volume of this tiling, but due to my loose knowledge of French, I cannot understand this text.


If you can tell me about an English translation of this text or if you can give me a reference that provides the same material in English, I would be very thankful.



English translation of M.-F. Vigneras "Arithmétique des algèbres de quaternions"
There might be an English translation of Vigneras book. If not, at least there is [Arithmetic of Hyperbolic 3-manifolds](http://www.springer.com/gb/book/9780387983868) by Colin McLachlan and Alan Reid. That book has information about quaternion algebras. 


I'll look into discussion of the fundamental volume of the tiling over $\mathbb{H}^3$.


In Chapter 11 they discuss [Tamagawa measure](https://mathoverflow.net/questions/111339/simple-tamagawa-number-calculations). If $A$ is a quaternion algebra, and $\mathcal{O} \subseteq A $ is a maximal order then we've got a formula for the volume: 


$$\text{Vol}\big( SL(2,\mathbb{C}) / \rho (\mathcal{O} ) \big)=
\frac{|\Delta\_k|^{3/2}\zeta\_k(2)\prod\_{\mathcal{P}\big|N(A)}\big( N(\mathcal{P})-1\big)}
{(4\pi^2)^{[k:\mathbb{Q}]-2}}$$


This volume is expressed in terms of invariants of the quaternion algebra, which are in the textbook. For example $k/\mathbb{Q}$ is the number field which was used to define $A$, and $\zeta\_k(2)$ is the Zeta function. There's another formula for volumes of $SL(2,\mathbb{R}) $ quotients as well. 


