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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 $ …

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model fo …

$S$ cannot be a 3-dimensional real hyperboloid unless $\psi$ is of the same form as $\phi$, though it can be a 2-dimensional real hyperboloid*. Excluding the possibilities achievable when $\psi$ is a …

As discussed in the comments, the contradiction is that $D$ isn't really the Dirichlet domain for $G$ centered at $i$. The red edge is in fact an edge of the domain.

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 …

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 $\mathcal{I}^3\subset\mathbb{R}^4$ 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 …

This is a follow-up question to a question from math.stackexchange: https://math.stackexchange.com/q/1436253/67563 Had it not been for the exchange there between myself and @Lee_Mosher in the comments …

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( …

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, …

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 realize there are a few different ways of going about proving this, depending on one's background, but there's a particular number theoretic aspect that I am just blanking on, and can't seem to find …

Let $\Gamma$ be a Kleinian group and let $\mathbb{H}^3$ be the upper half-space model for hyperbolic 3-space. Then $\mathbb{H}^3/\Gamma$ is an orientable hyperbolic 3-orbifold (with the group action d …

No. (See Ian Agol's comment.)