## Some pictures in the hyperbolic 3-space

The following are some pictures created as part of my semester project in the 2017-18 Fall semester at EPFL. The project was under the guidance of Prof. Maryna Viazoska. The project involved building the prerequisites and understanding this paper of Don Zagier.  Below, I will give some very brief details of the pictures and the project. The full write-up that I wrote for this project can be found here.

The hyperbolic 3-space is a Riemannian manifold. As a set, the Hyperbolic 3-space ${\mathbb{H}^3}$ can be visualized as ${\mathbb{R}^2 \times \mathbb{R}_{+}}$, given the metric tensor of the Riemannian manifold given by

$\displaystyle ds^2 = (dx^2+dy^2+dt^2)/t^2. \ \ \ \ \ (1)$

The following picture is of the set given by

$\displaystyle \{(x,y,t) \in \mathbb{R}^2\times \mathbb{R}_+\ | \ x^2+y^2+t^2 \ge 1, x\le \frac{1}{2} , y \le \frac{1}{2}, x+y \ge 0 \}. \ \ \ \ \ (2)$

This set is the fundamental region of the action of the group ${PSL(2,\mathbb{Z}[i])}$ as a discrete subgroup of ${PSL(2,\mathbb{C})}$, the group of isometries on the Hyperbolic 3-space, visualized as the upper half-space of ${\mathbb{R}^3}$ in the above picture.

Measuring volumes in Hyperbolic 3-space (volume measured according to the metric tensor) is done by utilizing the concept of ideal tetrahedra. A tetrahedron is said to be an ideal tetrahedron if it has at least one ideal vertex in the upper-half space model, and one vertex at ${\infty}$, and the projection of the tetrahedron from ${\infty}$ is a right-angled triangle on the ${xy}$ plane. And ideal vertex in the upper-half space model is a vertex which is of the form of two vertical planes (spheres containing infinity), whose axis of intersection is intersecting with the unit hemispherical geodesic perpendicularly (and therefore, at the apex point of the hemisphere.

The volume of such a tetrahedron is given by the following formula in the upcoming proposition, and an illustrative picture right after.

Proposition 1 Suppose ${T_{\alpha, \gamma}}$ is the ideal tetrahedra that has a dihedral angle (the angle between the two vertical planes) as ${\gamma}$ and has an acute angle (that is, the angle between the geodesic plane opposite to ${\gamma}$ and the unit geodesic hemisphere) given by ${\alpha}$. Then the following is true.

$\displaystyle \text{Vol}(T_{\alpha,\gamma}) = \frac{1}{4}(\mathcal{L}(\alpha+\gamma)+\mathcal{L}(\alpha-\gamma)+2\mathcal{L}(\frac{\pi}{2}-\alpha)). \ \ \ \ \ (3)$

Here ${\mathcal{L}}$ is the Lobachevski function, described by the equation

$\displaystyle \mathcal{L}(\theta)= - \int_{0}^{\theta} \log|2\sin(u)| du. \ \ \ \ \ (4)$

The proof of this proposition can be found in the write-up. Here is the picture of a sample ideal tetrahedron.

For more complicated tetrahedrons, one has to decompose it by writing it as a sum and difference of ideal tetrahedrons. Any tetrahedron can be seen as the difference of two tetrahedrons with one vertex at infinity (after applying suitable isometries), both of which can be decomposed into ideal tetrahedra as shown below.

I will very crudely mention the purpose of seeing these things in a write-up about the special values of Dedekind-zeta functions.

The project that I had done entails Zagier’s generalized expression for a particular special value of the Dedekind zeta function. The generalized expression was obtained by noting the appearance of the Dedekind zeta function in the expressions of the volumes of arithmetic hyperbolic 3-manifolds, and after some careful manipulation of these volumes on arrives at the following theorem.

Theorem 2 (Zagier, 1986) Let ${K}$ be an arbitrary number field. Suppose ${A(x)}$ is a real valued function given by

$\displaystyle A(x)=\int_0^x \frac{1}{1+t^2}\log{\left( \frac{4}{1+t^2} \right)}dt. \ \ \ \ \ (5)$

Then we have that the value of ${\zeta_K(2)}$ is given by an expression as below.

$\displaystyle \zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{D}}\sum_\nu c_\nu A(x_{\nu,1})A(x_{\nu,2})\dots A(x_{\nu,n}). \ \ \ \ \ (6)$

where the sum above is finite, ${D}$ is the discriminant of the number field, ${r}$ is the number of real places and ${s}$ is the number of complex places associated to the number field ${K}$. Moreover, the ${c_\nu}$ are rational and ${x_{\mu,j}}$ are real algebraic numbers.

Here is some motivation for studying this theorem. In 1734, Euler resolved the famous Basel problem which asked the following question:

What does the sum of the following infinite series converge to?

$\displaystyle \frac{1}{1^2}+ \frac{1}{2^2}+ \frac{1}{3^2}+ \frac{1}{4^2} \dots \ \ \ \ \ (7)$

The answer was found to be ${\frac{\pi^2}{6}}$ by Euler. This has the following consequence in modern terminology. We define the Riemann-zeta function for ${1 to be the function

$\displaystyle \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}. \ \ \ \ \ (8)$

So rephrasing the Basel’s problem simply amounts to saying that ${\zeta(2)=\frac{\pi^2}{6}}$. We will consider the generalization of the identity of Equation (8) in the case of the Dedekind zeta function.

The Dedekind zeta ${\zeta_K}$ associated to the number field ${K}$ is defined as the following for a number field ${K}$ and for ${1.

$\displaystyle \zeta_{K}(s)=\sum_{ \mathcal{I} \text{ is a non-zero integer ideal of } K} N(\mathcal{I})^{-s}. \ \ \ \ \ (9)$

The Dedekind zeta function is an important function that has important connections in number theory. Putting ${s=2}$ and ${K}$ as the simplest number field ${\mathbb{Q}}$, the above equation turns into Equation (7). Zagier’s theorem is a very nice generalization of the above identity of Euler’s.

In the report, I have spent a considerable length justifying the appearance of the Dedekind zeta function in the volume formula of hyperbolic 3-manifolds.

I believe that this project was by far the most intense reading project I had ever taken.

## Deck transformations, revisted

A long time ago, I had animated the deck transformations of the covering space of a pair of circles. I’m talking about this blog post.

A lot of time has passed and I couldn’t help but rethink it in the light of some insights that I later had after the post. The deck transformations that I had created in my first attempt are quite unnatural. The reason is simply that a Cayley graph of a free group of two generators cannot be embedded inside ${\mathbb{R}^2}$ with the geometry intact. Therefore, any attempt to visualize the Cayley graph in that setting means forgetting about the geometry.

However, there is one setting in which the Cayley graph can naturally exist, without lying about its geometry. And that place is the hyperbolic 2-space (or hyperbolic n-space)! I was inspired by this idea when I found about this project called Walrus. This program is a way to visualize exponentially growing graphs using tools of hyperbolic geometry. The Cayley graph discussed in the previous blog post is also an exponentially growing graph.

Earlier, we had considered the universal covering space of the following topological space (think of it as a subspace of ${\mathbb{R}^2}$).

The covering space looked like the following. It is a graph, better expressed with a bunch of line segments quotiented away at the nodes that connect them.

Notice that the above graph is the same as the following graph. All the vertices are tetravalent like how they should be. As some of you may be able to guess, this graph is actually embedded inside the Poincare disc, with the line segments now being the Hyperbolic geodesics of the negative curvature space.

For clearer understanding, let us glance over what I mean. As a set, the Poincare disc ${D}$ is given by

$\displaystyle D= \{ z \in \mathbb{C} \ | \ |z| < 1\}. \ \ \ \ \ (1)$

However, I would be kidding you if I told you that this is it. The Poincare disc is a metric space with the following metric. For two points ${z_1,z_2 \in D}$, the distance ${d(z_1,z_2)}$ between them is given by

$\displaystyle \cosh(d(z_1,z_2)) = 1 + 2\frac{|z_1-z_2|^2}{(1-|z_1|^2)(1-|z_2|^2)} . \ \ \ \ \ (2)$

Of course, this bizarre formula is not very telling. It’s not even clear why this is a metric (for instance, why does it satisfy the triangle inequality?). Questions like these, and why this metric space is interesting will require a lot of discussion and we’ll not get into it. Interested readers can find this in plenty of books. To get a short introduction, you can watch this numberphile video.

The group of isometries (set of distance preserving homeomorphisms ${D \rightarrow D}$) of the space ${D}$ also happen to have a very neat and natural description. It is the group ${PSU(1,1)}$ which is the group defined as

$\displaystyle PSU(1,1):= \{ \left( \substack{a\ b \\ \bar{b}\ \bar{a}} \right)\ ,|a|^2-|b|^2=1 , (a,b,c,d) \in \mathbb{C}^4 \} / \{ \pm 1\} . \ \ \ \ \ (3)$

The action of this group on ${D}$ is via

$\displaystyle z \mapsto \frac{az+b}{cz+d}. \ \ \ \ \ (4)$

Enthusiastic readers of my blog (if there are any) will be able to relate the above formula to the modular group action in this blog post. They’re both examples of fractional linear transformations. In fact, the group ${PSU(1,1)}$ is isomorphic to the group ${PSL(2,\mathbb{C})}$ via a conjugation by some matrix (there are many such). Talking about this will lead us to the upper-half plane model of Hyperbolic geometry, the discussion of which we will save for some other time.

I am interested in two particular matrices:

$\displaystyle A:= \pm \Big{(} \substack{\sqrt{2}\ \ 1 \\ 1 \ \ \sqrt{2}}\Big{)} ,\ B:= \pm \Big{(} \substack{\sqrt{2}\ \ i \\ -i\ \ \sqrt{2}} \Big{)}. \ \ \ \ \ (5)$

These matrices are carefully chosen, so that the following happens. Here is the action of ${A}$ on the picture produced previously.

Here is the action of ${B}$.

How were those specific matrices calculated? It’s enough for me to show this picture to you perhaps. This is the fundamental region of the group ${\langle A,B \rangle \subset PSU(1,1)}$, which is the subgroup of ${PSU(1,1)}$ generated by the matrices ${A}$ and ${B}$.Generating the entire tiling using the above fundamental region gives us the following pictures (the pointy edges in the four corners are something that is unnatural. It is because I could only calculate finitely many tiles. Those pointy corners would look denser, if it were calculated by a computer that had more time).

Here are the actions of ${A}$ and ${B}$ on the above picture.

For clarity, I will also demonstrate the Cayley tree with the tiling appearing faintly in the background.

This group ${\langle A, B\rangle}$ is, in fact, isomorphic to the free group of two generators, as you would expect. It must be isomorphic to the group of deck transformations of the covering space of ${B}$. Another direct way to prove it is to use the Ping-Pong lemma.

Here are some more trippy visuals of the action of ${A}$ and ${B}$.

Although in the previous post I had made a visualization of the covering map itself, I don’t think there is a natural way that map can be visualized using hyperbolic isometries, without which the purity of the Hyperbolic space will be ruined. If any of my readers gets an insight of having a way to map the Cayley tree to the wedge of two circles in a natural way, I would be excited to listen to your opinions in the comments.

One important discussion here may be the fact that ${\frac{D}{\langle A, B\rangle}}$, which is the topological space of orbits of ${D}$ under the action of ${\langle A, B\rangle}$, is homeomorphic to a torus with a point removed. A torus with a point removed has the same homotopy equivalence (and hence the same fundamental group) as that of a wedge of two circles (How? I wouldn’t ruin it for you). Do you smell a connection? It’s an interesting bedtime thought, trying to put this picture together. Someday maybe in a future project, I will try to make a visualization about it.

Here’s an exercise for you. Can you guess which element of ${\langle A, B\rangle}$ does the following transformation correspond to?