@johncarlosbaez@mathstodon.xyz The figure eight knot group Γ is generated by the Mobius transformations z-> z+ω, z-> 1/(1+z), acting on the upper half-space model of hyperbolic space, with coordinates (z,t), z in C, t>0, and where ω is a third root of unity (-1+sqrt(-3))/2. There is a cusp H (or horoball) centered at infinity, which is the set of points t> 1. Looking at the image of H under the group Γ, one gets balls tangent to the boundary t=0 and of height 1, centered at the lattice Z+Zω, as well as horoballs of various heights tangent to Q(ω). This is what is shown in the image. They are tangent to H, and so are the largest horoball whose interior is disjoint from all of its images. The volume of the image of H in the quotient is sqrt(3), and the volume of the quotient is 2.0298. This ratio is a rational multiple of their value (and I think I had it flipped, 1/ cusp density, so volume / cusp volume).
