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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 …

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 arithmet …

When I say manifold below, I mean a complete orientable finite-volume hyperbolic $3$-manifold and when I say subsurface, I mean immersed closed totally geodesic subsurface. Whenever a manifold has a …

Let $X$ be a complete finite-volume orientable hyperbolic $3$-manifold, and let $\Gamma$ be a Kleinian representation of $\pi_1(X)$. Let $K\Gamma:=\mathbb{Q}\big(\{\mathrm{tr}\mid\gamma\in\Gamma\}\big …

The answer to this follows easily from section 9.5 of Machlachlan and Reid -- which somehow I managed not to see every other time I checked the book! The thing stated in my question is not true, but t …

Let $\mathcal{I}^3\subset\mathbb{R}^4$ be the standard hyperboloid model for hyperbolic $3$-space and consider the usual $\mathrm{SO}(3,1)$ action of $\mathrm{PSL}(2,\mathbb{C})$ on $\mathcal{I}^3$. G …

I think made this a little too complicated, and I don't think there is a lot of interest in the topic in the first place. So let me just add something more conclusive for the sake of resolving the pos …

UPDATE - Feb. 9, 2017: The original title of this post was "The $\text{isometry}^+$ group of hyperbolic $n$-space as $\mathrm{PSL}_2$ of a Clifford group." The original question, which appears below, …

After reading a bunch, I've noticed there is much disagreement about how to define $\mathrm{PSL}_2(\mathbb{H})$. The conditions one might use to define this are best expressed in terms of the Cliffor …

Edit: In my original post I failed to require the group to be a manifold group. The answer below from @BenLinowitz works in that case. I am really interested though in when the group is torsion-free, …

SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting: the minimal polynomial of the field over $\mathbb{Q}$, and a decimal ap …

I am interested in realizing commensurability classes of hyperbolic $3$-manifolds whose quaternion algebra (note: not invariant quaternion algebra) is isomorphic to one of the form $\Big(\frac{a,b}{F( …

The comments from @IanAgol above lead to an affirmative answer in the compact case. This paper gives an affirmative answer in the non-compact case: http://www.math.umt.edu/chesebro/AIMCLC.pdf

I am interested in when the trace field of a knot complement has the form $F(\sqrt{-d})$ for $F\subset\mathbb{R}$ and $d\in F^+$ (squarefree). Does this occur for infinitely many choices of pairs $(F, …

A matrix $X=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{PSL}_2(\mathbb{C})$ acts isometrically on the upper half-space model $\mathbb{H}^3$ via isometric extension of the Mobius transformation on $ …