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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.

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    $\begingroup$ Did you try John Voight's book in progress math.dartmouth.edu/~jvoight/quat.html ? $\endgroup$ Oct 4, 2017 at 19:05
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    $\begingroup$ In addition to any other answers, it is fair to say that you should probably learn to read mathematical French. (This is unavoidable for me as a representation theorist, but I imagine it's true in many disciplines.) It's way easier to learn the mathematical version than even the most basic conversational version of the language. $\endgroup$
    – LSpice
    Oct 4, 2017 at 20:12
  • $\begingroup$ @PiotrAchinger Thank you! It seems to be a useful reference. $\endgroup$ Oct 4, 2017 at 22:14
  • $\begingroup$ @LSpice Thanks for the suggestion. I am in the pursuit of learning French and your suggestion certainly motivates me. $\endgroup$ Oct 4, 2017 at 22:15
  • $\begingroup$ yumpu.com/en/document/view/7921447/… $\endgroup$ Oct 5, 2017 at 4:41

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There might be an English translation of Vigneras book. If not, at least there is Arithmetic of Hyperbolic 3-manifolds 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. 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.

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  • $\begingroup$ This is not my specialty. I am using this answer as learning opportunity. Also interested in the use of $\zeta_F (2)$ for number fields. $\endgroup$ Oct 4, 2017 at 20:18
  • $\begingroup$ Thanks a lot for the information. That was certainly helpful. $\endgroup$ Oct 4, 2017 at 21:45

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