Joint IAS/Princeton University Number Theory Seminar
From counting Markoff triples to Apollonian packings; a path via elliptic K3 surfaces and their ample cones
The number of integer Markoff triples below a given bound has a nice asymptotic formula with an exponent of growth of 2. The exponent of growth for the Markoff-Hurwitz equations, on the other hand, is in general not an integer. Certain elliptic K3 surfaces can be thought of as smooth generalizations of the Markoff surface. Like the Markoff surface, their group of automorphisms is discrete and infinite. One can therefore investigate the growth rate of a rational point (or curve) on the surface under the action of the group. The exponent of growth is sometimes an integer, and sometimes not. When it is not, it can be thought* of as the Hausdorff dimension of a fractal associated to the ample cone. For some of them, that fractal is the residual set for the Apollonian packing. (*Partially known, otherwise conjectured.)