# Some general properties of arithmetic groups of simplest type

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?

• There is a standard reference in English for arithmetic groups in three dimensional hyperbolic space (and more), which is the book "The arithmetic of hyperbolic three--manifolds" by macLachlan and Reid. It includes proofs for your statements 1--3 (1 only in SL_2(C), but you can find a general proof in any basic course on arithmetic groups). The review is here: ams.org/mathscinet-getitem?mr=1937957. – Jean Raimbault Feb 26 '15 at 16:23
• Actually I've read much of that book, but had not read the last few chapters which, looking again, do seem to address some of this. Are you referring to Ch. 10? They do not mention Vinberg's "groups of simplest type" by name (correct me if I'm wrong about that), and their definition of the field of definition is not equivalent to Vinberg's -- for instance for them the inv. trace field is the least such field and that is not real. But they cover an equivalent formulation? Regardless I will take a closer look. – j0equ1nn Feb 27 '15 at 7:12
• As you more or less said in your question, the field of definition in Vinberg's sense (the field over which the quadratic form is defined) corresponds to the real subfield (at the unique complex place) of the trace field of the quaternion algebra, when the index of this subfield is 2 (otherwise the isomorphism SL_2(C) \to SO(3,1) you get from 10.3 in macLachlan--Reid is not defined over the "correct" number field, so you don't get a lattice "of simplest type"). – Jean Raimbault Feb 27 '15 at 12:44
• I know what you mean simplest type requiring $[k\Gamma:k\Gamma\cap\mathbb{R}]=2$. I was unaware though that M/R's $\mathrm{SL}(2,\mathbb{C})\overset{\cong}{\rightarrow}\mathrm{SO}(3,1)$ corresponds with Vinberg's construction in that case. That is very helpful, and so I'll be putting in some time with Ch.10 this week. (I know I tagged this a reference-request but I do want to understand the details as well.) Would you consider rewriting your remarks as an answer? It seems the best answer I'm gonna get, I just have to do some homework! – j0equ1nn Feb 27 '15 at 23:45

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 $234.40 so I'm looking at the Google preview. Searching the word "quaternion" there yields no results. This use of quaternions does not apply in arbitrary dimensions, and that is the author's topic. The use of quaternion algebras in characterizing arithmeticity is described in the Hilden-Lozano-Montesinos chapter of Topology 90. There's also a good explanation of it, with examples, in Kate Peterson's article "Arithmetic groups and Lehmer's conjecture." I have seen noting in print proving the 3rd statement, but it's a good exercise to work it out for oneself. Neumann and Reid's paper "Arithmetic of Hyperbolic Manifolds" is also a good reference for citation. For example, (1-3) also follow from the arguments in that paper. This is of course not to try to steer the OP away from Jean Raimbault's suggestion of Maclachlan and Reid's book. The book provides a thorough treatment of the subject matter and ends each chapter with a background section, making it a great starting point for any reference request in this area. • I'm familiar with that paper and I like it too but... The paper makes no mention of groups of simplest type. It does not even mention lattices in$\mathbb{R}^4\$ or quadratic forms, never mind groups of motions preserving them. It is Vinberg's particular definition which is of interest to me. Would you know of some lead for a way to see Vinberg's definition of "simplest type" as equivalent to one of the definitions or characterizations in this paper? Where do they restrict attention to this particular type of arithmetic group? – j0equ1nn Feb 27 '15 at 7:38