TITLE: Kissing Number of Spheres in Non-Euclidean Geometry
QUESTION [7 upvotes]: There has been much work done on the kissing number problem (of determining the greatest number of congruent spheres which can touch a single sphere in a packing) in Euclidean space for dimensions $1$ to $24$ and even some asymptotic work. My question is whether or not the kissing number problem has been studied in Non-Euclidean geometries such as the other seven Thurston Geometries $\mathbb{S}^2 \times \mathbb{R}, \mathbb{H}^2 \times \mathbb{R}, \mathbb{S}^3, \mathbb{H}^3, \widetilde{SL_{2}(\mathbb{R})}$, Nilgeometry, or Solvgeometry.
Any references or ideas about this topic would be very interesting!

REPLY [5 votes]: In one sense, which you are free to see as unimportant, the kissing number in the standard hyperbolic plane is unbounded, as it increases with increasing radii of the disks. One of the Laws of Cosines says
$$ \cos \alpha = 1 - \frac{1}{1 + \cosh a},  $$
where $a$ is the side length of an equilateral triangle and $\alpha$ is the vertex angle. As $a$ increases without bound, $\alpha$ decreases without nonzero lower bound. So, with discs of large enough radius, we can place as many disjoint-interior disks as we like around a given central one. The equilateral triangle has vertices at the centers of three mutually tangent circles. 
I'm just saying.