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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.

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.

There might be an English translation of Vigneras book. If not, at least three 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.

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.

There might be an English translation of Vigneras book. If not, at least three 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:

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

There might be an English translation of Vigneras book. If not, at least three 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.