Lattice packings of the hyperbolic plane

The lattice sphere packing problem in Euclidean space asks for the densest possible lattice packing of spheres in R^n, for a(ny) fixed n.  Here a lattice is the set of integral linear combinations of a fixed basis, and a lattice packing is the set of translates of a fixed sphere by all lattice elements, assuming this sphere is small enough that it doesn't overlap any of those translates.  We can translate the lattice sphere packing problem to the setting of hyperbolic space by translating the notion of "lattice".  It becomes a group of isometries (i.e. rigid motions, analogous to translations) acting discontinuously (analogous to linear independence of the Euclidean basis vectors) with finite-volume quotient (analogous to the Euclidean basis vectors spanning R^n).

I will present a solution to the resulting lattice sphere packing problem for the hyperbolic plane.  The hyperbolic plane's quotient by the action of a lattice of isometries can have many different topological types, and there is a different maximal packing density for each type.  For lattices with compact quotient, the maximal-density lattice packings have maximal density among all (not just lattice) packings of the hyperbolic plane, but this does not hold for lattices with non-compact quotient.  This is notable as it contrasts all dimensions where maximal densities are known for both lattice packings and arbitrary packings of Euclidean space (2, 3, 8, and 24).

I will also discuss some of the machinery used in the proof of this result.  In particular, I will talk a bit about the "meshing" and "skinny triangles" problems in the context of packings.  This is an alpha version of a talk intended for a general audience.