Generating hyperbolic tessellations?

Question

This might be a naive question, but I’d like to seek clarification to avoid confusion. I’m interested in describing from a group-theoretic perspective a crystal structure in hyperbolic space from tessellations of the hyperbolic space.

To provide some motivation, let’s first examine the Euclidean case. Consider $\mathbb Z^n$ as a subgroup of the Euclidean group $\mathbb E(n)$. Note that $\mathbb Z^n$ is a lattice of maximal rank in $\mathbb E(n)$. The action of $\mathbb Z^n$ is smooth, free and proper. The orbit space $\mathbb Z^n \cdot (0,\cdots,0)$ generates a uniform lattice in $\mathbb R^n$. The quotient space $\mathbb R^n / \mathbb Z^n$ is compact. Let $G$ denote the point group of $\mathbb Z^n \cdot (0,\cdots,0)$, or the group of symmetries of $\mathbb Z^n \cdot (0,\cdots,0)$ modulo translations.

I would like to generalize this situation to the case of tessellation of hyperbolic space. Here is my attempt. Consider $\mathbb H^n$ for $n=2,3$. Let $\Gamma \subseteq \text{Isom}(\mathbb H^n)$ be a discrete, maximal, and co-compact subgroup (lattice?) such that $\Gamma$ acts on $\mathbb H^n$ smoothly, freely and properly. Choose a $x \in \mathbb H^n$ and identify the orbit space $\Gamma \cdot x$ with a lattice in $\mathbb H^n$.

Is this the correct way to think about generating tessellations of $\mathbb H^n$? Can this method be used to generate various uniform ($\{p,q\}$ tessellations etc.) and non-uniform tessellations? If so, I have a few questions:

1. Can we pick $x \in \mathbb H^n$ arbitrarily? Will every orbit space be diffeomorphic? I am not so sure.
2. How does one generate a $\{p,q\}$ tessellation of $\mathbb H^2$ using this scheme? I can find any reference that pin points the choice of the Fuchsian group.
3. How does one identify the analog of the point group in the hyperbolic case? Do we need to do it? Naively, "rotational symmetries" might already be built in the choice of $\Gamma$. This is not the Euclidean case where we chose $\mathbb Z^n$. I am not sure.
4. Let $n=2$. Since $\mathbb H^n/\Gamma$ is compact, $\Gamma$ doesn't contain any parabolic elements. Can all tessellations be generated by Fuchsian groups that are surface groups? Probably not. I think we'll only get tessellations by $4g$-ons in the surface group case. But I’ll leave this question here.

Answer 1

I started to write this as a comment, but it is clearly too long, so I will post this as an answer.

1. You are mistaken about point-groups in the case of $\mathbb Z^2$: This group depends on which lattice you take. Generically, it is only $\mathbb Z_2$; for "typical" rectangular lattice the point-group of $\mathbb Z_2\times \mathbb Z_2$; for "square" lattice, it is indeed $D_4$. However, the largest possible point-group is the dihedral group $D_6$ (in the language of Coxeter groups, this is the type $\widetilde{G}_2$).

2. As I noted in a comment to your other question, you absolutely cannot identify identify $\mathbb H^n/\Gamma$ with the unit cell (a fundamental domain). This is a common mistake. The space $\mathbb H^n/\Gamma$ is a quotient of a fundamental domain. Only occasionally, there is a homeomorphism between the two (the case of reflection groups).

3. What you are also missing, if you want to construct a tessellation of $\mathbb H^n$, is a choice of a fundamental domain. Given $x$, the most common choice is the Dirichlet fundamental domain $D$. Taking $\gamma$-translates of $D$ ($\gamma\in \Gamma$) defines a tessellation of $\mathbb H^n$.

4. $(p,q)$-tessellations do not correspond to maximal lattices acting freely (as you assume in your post). They correspond to finite index subgroups $\Gamma$ in $(2, p, q)$-triangle Coxeter groups $\widehat\Gamma$ acting on the hyperbolic plane. This is because the tiles of such tessellations have extra dihedral groups of symmetry, $D_{2p}$. The group $\widehat\Gamma$ splits a semidirect product $$ \widehat\Gamma\cong \Gamma \rtimes D_{2p}. $$ The group $\widehat\Gamma$ is the full symmetry group of the $(p,q)$-tessellation. (When $p=q$ there is even larger group containing $\widehat\Gamma$ as an index 2 subgroup, otherwise, $\widehat\Gamma$ is maximal.) The subgroup $\Gamma$ itself is also a Coxeter group, the one with the presentation $$ \langle x_1,...,x_p| x_i^2=1, (x_i x_{i+1})^q=1, i=1,...,p\rangle $$ where $i$ is taken moduli $p$, i.e. $p+1=1$.

5. Unlike the Euclidean case, there is no "canonical" notion of point-groups and people avoid using this notion. You can talk about stabilizers of various points in the hyperbolic space. Or, you can regard these groups as extra symmetry groups, comparing to your original lattice (like $D_{2p}$ in the example above).

I will add more details later on (you are asking too many questions in one post).