Does every hyperbolic space admit a vertex-transitive 'grid'?

Question

This isn't exactly a question about tiling. It's closely related enough that I though the tag was justified but it's not closely related enough that I can figure out how to apply the existing literature to this question.

Let $\mathbf{H}^n$ be the $n$-dimensional hyperbolic space with its standard Riemannian metric. Does there always exist a set $X\subseteq \mathbf{H}^n$ with the following properties?

• $X$ is uniformly discrete, i.e. there is an $\varepsilon > 0$ such that for any distinct $x,y\in X$, $d(x,y) > \varepsilon$.
• $X$ uniformly covers $\mathbf{H}^n$, i.e. there is a $\delta > 0$ such that for any $x\in \mathbf{H}^n$ there is a $y\in X$ such that $d(x,y) < \delta$.
• $X$ is vertex-transitive, i.e. for any $x,y\in X$ there is a isometry $f:\mathbf{H}^n\rightarrow \mathbf{H}^n$ such that $y=f(x)$ and $f(X)=X$.

The only relevant result I have found is on this Wikipedia article, which mentions that "There are no compact regular tessellations of hyperbolic space of dimension 5 or higher and no paracompact regular tessellations in hyperbolic space of dimension 6 or higher." But it seems like that's requiring more regularity than I am.

Answer 1

Your desired set $X \subset \mathbb H^n$ can be constructed if you have a subgroup $\Gamma$ of the isometry group $\text{Isom}(\mathbb H^n)$ which satisfies the following properties:

1. ($\Gamma$ is discrete) For each $x \in \mathbb H^n$ the orbit $\Gamma \cdot x = \{\gamma(x) \mid \gamma \in \Gamma\}$ is a discrete subset of $\mathbb H^n$
2. ($\Gamma$ is cocompact) There exists a compact set $D \subset \mathbb H^n$ such that $\Gamma \cdot D = \{\gamma(x) \mid \gamma \in \Gamma, x \in D\}$ is equal to the whole of $\mathbb H^n$.

Once you have that, then for your set $X$ you can simply take an orbit $X = \Gamma \cdot x$, for any $x \in \mathbb H^n$.

A discrete cocompact subgroup of $\text{Isom}(\mathbb H^n)$ is also called a cocompact lattice in $\text{Isom}(\mathbb H^n)$ (there are also noncocompact lattices, but I won't go into that).

Cocompact lattices can be constructed in $\text{Isom}(\mathbb H^n)$ for every $n$, but the constructions in high dimensions are not easy. I notice that the wikipedia pages don't seem to contain too much information about constructions, more just about the general theory. One exception is this page although the existence statement is set in a very general context. The connection on that page with hyperbolic space is here where they discuss $SO(n,1)$, which is matrix representation of $\text{Isom}(\mathbb H^n)$.