I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of arithmetic groups termed *simplest type*, first defined in Chapter 6 of Vinberg and Shvartzman's *Discrete Groups of Motions of Spaces of Constant Curvature* (1993). As I said I'm mostly concerned with $n=3$ but let me give the more general definition as in the reference.


A group is called an *arithmetic discrete group of motions of simplest type* with *field of definiton* $K$ if it is commensurable to a group $\Gamma$ which arises in the following way.
Let $K\subset\mathbb{R}$ be a totally real number field with ring of integers $R\_K$.
Let $f(x)=\sum\_{i,j=0}^na\_{i,j}x\_ix\_j$ $(a\_{i,j}=a\_{j,i}\in K)$ be a non-degenerate quadratic form of signature $(3,1)$, such that for any non-identity embedding $\sigma:K\hookrightarrow\mathbb{R}$, the quadratic form $f^{\sigma}(x)=\sum\_{i,j=0}^na\_{i,j}^{\sigma}x\_ix\_j$ is positive definite.
Let $C$ be the cone $\{x\in\mathbb{R}^{n+1}\mid f(x)<0\}$, and let $L\subset K^{n+1}$ be a complete lattice (with basis in $R\_K$).
$\Gamma$ is a group that preserves $f$, preserves $L$, and sends each connected component of $C$ onto itself. 


The authors make some statements without proof (listed below). There are references given but these are not accessible (or are in Russian). I've discovered a way of proving $2\frac{1}{2}$ of the statements, which I'm hoping to complete and generalize beyond the simplest type setting. But I want to make sure my idea is different enough from the standard way, and I want to check on the conventional use of the term *field of definition* when generalized to arithmetic Kleinian groups (not necessarily of simplest type).


Those statements:


1. The group is non-cocompact if and only if $K=\mathbb{Q}$ and $f$ nontrivially represents zero in $\mathbb{Q}$.
2. The Bianchi group $\mathsf{PSL}\_2(\mathcal{O}\_d)$ is commensurable to a group $\Gamma$ as above where $f=x\_1x\_2+x\_3^2+dx\_4^2$ and $L=\mathbb{Z}^4$.
3. A group derived from a quaternion algebra over a field $F$ is of simplest type if and only if $F$ is an imaginary quadratic extension of a real number field.


The first two items I'm able to do plus get what $L$ is precisely (not up to commensurability), so I'm mainly interested in what methods of proof the references (or standard methods) use. The third item I am still working out, trying to find explicit forms and lattices for cocompact groups, so I'd be interested in more detail there, but would be content to get some idea of the standard methods in proving the statement.


Lastly, I've noticed in some papers, when discussing a *field of definition* for an arithmetic Kleinian group, it is often not totally real, and I think the convention is for it to be the field $F$ that the corresponding quaternion algebra is over (which is necessarily not real). So, is it true that Vinberg's $K$ is always $F\cap\mathbb{R}$? And if so what is the obstruction to $F\cap\mathbb{R}$ being useful for non-simplest-type groups?



Some general properties of arithmetic groups of simplest type
The book *Conformal Geometry of Discrete Groups and Manifolds* by Boris N. Apanasov contains a detailed description of what it means for a group to be arithmetic, and why the definition manifests as it does for topology applications. He includes examples that illustrate the ideas behind statements 1 and 2, and discusses Vinberg's "simplest type" groups separately, calling them "real arithmetic" groups. He does not address the characterization using quaternion algebras, as far as I can tell. See, the book is $234.40 so I'm looking at the Google preview. Searching the word "quaternion" there yields no results. This use of quaternions does not apply in arbitrary dimensions, and that is the author's topic.


The use of quaternion algebras in characterizing arithmeticity is described in the Hilden-Lozano-Montesinos chapter of *Topology 90*. There's also a good explanation of it, with examples, in Kate Peterson's article "Arithmetic groups and Lehmer's conjecture." I have seen noting in print proving the 3rd statement, but it's a good exercise to work it out for oneself.


