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The upper half-plane model for real hyperbolic $2$-space is typically defined as $$\big\{p=x_1+x_2i\in\mathbb{C}\mid x_1\in\mathbb{R}, x_2\in\mathbb{R}^+\big\}, \quad dp=\sqrt{\frac{x_1^2+x_2^2}{x_2^2}}.$$ Similarly, the upper half-space model for real hyperbolic $3$-space is typically defined as $$\big\{p=(x_1,x_2)\mid x_1\in\mathbb{C},x_2\in\mathbb{R}^+\big\},\quad dp=\sqrt{\frac{|x_1|^2+x_2^2}{x_2^2}}.$$ A less common way of writing the latter is to write $p=x_1+x_2i+x_3j\in\mathbb{H}$ where $\mathbb{H}$ is Hamilton's quaternions, then require that $x_3>0$. This is nice because there is an analog of the Möbius action on $\mathbb{C}\cup\{\infty\}$ (the boundary of the hyperbolic half-plane) to this subspace of $\mathbb{H}$, which agrees with the usual isometric extension used in the $\mathbb{C}\times\mathbb{R}^+$ model (see Ratcliffe).

This is cool, but what happens when we go up another dimension? Well, in a lesser known construction, the upper half-space model for real hyperbolic $4$-space can be defined as $$\big\{p=x_1+x_2i+x_3j+x_4k\in\mathbb{H}\mid x_1\in\mathbb{R}^+\big\}, \quad dp=\sqrt\frac{x_1^2+x_2^2+x_3^2+x_4^2}{x_1^2}.$$ Here there is another analogous theory extending the usual use of modular forms from the lower dimensions (see Möbius transformations and the Poincaré distance in the quaternionic setting by Bisi and Gentile).

What bothers me is the inconsistency in the pattern as we climb through dimensions. In 2D we have $i\mathbb{R}^+$ for the half-axis. Then in 3D, we put $i\mathbb{R}$ on the boundary and use $j\mathbb{R}^+$ for the half-axis (at this point, noticing we could just written $i$ as $j$ in the 2D setting and everything works but invites confusion). Next in 4D, we change our minds again and use the real part of the algebra for the half axis. I would prefer to have done that all along. After all it is a common topic of interest to restrict a group action to a lower dimensional space, and study lower-dimensional submanifolds, etc.

So my question is, would we lose anything if we instead defined hyperbolic $2$ and $3$-space respectively as: $$\big\{p=x_1+ix_2\mid x_1\in\mathbb{R}^+\big\}, \quad dp=\sqrt{\frac{x_1^2+x_2^2}{x_1^2}}$$ and $$\big\{p=x_1+x_2i+x_3j\mid x_1\in\mathbb{R}^+\big\} \quad dp=\sqrt{\frac{x_1^2+x_2^2+x_3^2}{x_1^2}}$$ where $i$ and $j$ behave just as they do in $\mathbb{H}$? By "lose anything," I mean, would Möbius transformations work the same way? Would we hit any awkwardness applying the theory of modular forms to study hyperbolic space? Any other difficulty that we would encounter, in exchange for the niceness I've described?

We would gain, not only consistency through the dimensions, but clear implications for how to continue using Clifford algebras on higher dimensional spaces.


Added Feb. 14, 2016: The Möbius transformations break down in this approach. We can see this with an easy example. Let's define $\mathcal{H}^2:=\big\{z\in\mathbb{C}\mid\Re(z)>0\big\}$ and let $m=\begin{pmatrix}1&-1\\0&1\end{pmatrix}$. Then $m\in\mathrm{PSL}_2(\mathbb{R})$ and $1\in\mathcal{H}^2$, but $m(p)=\frac{1\cdot1-1}{0\cdot1-0}=0$ is not.

This calls a few other things into question about this post, it may need to be thought out better. (See discussion below with @MvG.)

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As an abstract concept, it doesn't really matter which component you restrict to the positive domain. You are essentially dealing with a vector space, and could define everything there without resorting to the algebraic structure, i.e. multiplication and division in the complex numbers. But using the complex structure may be used to express some concepts, and if you do those concepts may require similar cosmetic adaptations. For example, people may know that in 2d the set of all orientation-preserving isometries is equal to the projective transformations on the real line, and can easily be described by a homogeneous quadruple of real numbers. If you pick the imaginary axis instead, that form breaks down slightly and you need some imaginary units to fix this again. Nothing dramatic, you still have the same set of parameters in essence, but visually this looks slightly different.

So on the whole I'd you'd mostly loose the ability to blindly rely on established conventions. When changing the convention but interacting with literature that is using the original convention, you have to be aware of the distinction and be prepared to translate between conventions as needed.

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  • $\begingroup$ I just noticed something that easily shows a problem with picking $\mathbb{R}^+$ as the half-axis in $\mathcal{H}^2$. I think this answer then is too general, because this is specifically what I was talking about -- the use of Möbius transformations to get isometries of hyperbolic space. It seems there is something special (algebraically) about what happens using an imaginary half-axis that is beyond just changing conventions. $\endgroup$ – j0equ1nn Feb 14 '17 at 5:52
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    $\begingroup$ @j0equ1nn: That's what I meant when I wrote about isometries being projective transformations of the real line by convention. With $\operatorname{Re}(z)>0$ as the hyperbolic plane, isometries would be of the form $$\begin{pmatrix}a&ib\\ic&d\end{pmatrix}\qquad\text{with }a,b,c,d\in\mathbb R$$ i.e. you'd have purely imaginary entries off the diagonal. Or you multiply the matrix by $i$, then you have purely imaginary entries on the diagonal, and real off the diagonal. $\endgroup$ – MvG Feb 14 '17 at 6:57
  • $\begingroup$ Okay sure, but I don't think this sets us up for higher-dimensional analogies such as action on $\mathbb{R}^+\otimes\mathbb{H}_0$, does it? In fact, I am now doubting the sources claiming they have a linear-fractional transformation on that space -- they never actually write it. Also I think the generalization with Clifford algebras goes differently than how I wrote it, and the half axis is the next vector axis of the next Clifford algebra, consistent with the low dimensions. There's an awesome paper by Maclachlan, Waterman and Wielenberg from 1989 that does this in $\mathcal{H}^4$. $\endgroup$ – j0equ1nn Feb 14 '17 at 7:28
  • $\begingroup$ I honestly haven't worked out the higher-dimensional isometries in half-space models. Personally I guess I'd use Beltrami-Klein for higher dimensions in most cases. Or I'd formulate this using Lie sphere geometry, to have a decent representation of hyperbolic hyperplanes. But if you use some model over e.g. the quaternions, then exchanging the real axis with one of the imaginary ones should be easy, and therefore translating transformations between conventions should be easy as well. If you give me a transformation in one world, I can give you the result in the other, I guess. $\endgroup$ – MvG Feb 14 '17 at 7:38
  • $\begingroup$ Yeah, most sources prefer other models, especially Beltrame-Klein. I'm looking for something that lets me do hyperbolic geometry in $n$ dimensions using techniques like we have in geometric topology on hyp $3$-orbifolds. But I think I have tracked down most sources on this and what I'm looking for is mostly open -- making this a crappy math.stackexchange post (and one that probably would not get answered on mathoverflow). So I might as well accept your answer, I appreciate your sharing your insight with me. $\endgroup$ – j0equ1nn Feb 14 '17 at 7:45

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