From counting Markoff triples to Apollonian packings; a path via elliptic K3 surfaces and their ample cones
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 aÌýnice asymptotic formula with an exponent of growth of 2.Ìý The exponent ofÌýgrowth for the Markoff-Hurwitz equations, on the other hand, is in generalnot an integer.Ìý Certain elliptic K3 surfaces can be thought of as smoothÌýgeneralizations of the Markoff surface.Ìý Like the Markoff surface, theirÌýgroup of automorphisms is discrete and infinite.Ìý One can thereforeÌýinvestigate the growth rate of a rational point (or curve) on the surfaceÌýunder the action of the group.Ìý The exponent of growth is sometimes anÌýinteger, and sometimes not.Ìý When it is not, it can be thought* of as theÌýHausdorff dimension of a fractal associated to the ample cone.Ìý For some ofÌýthem, that fractal is the residual set for the Apollonian packing.Ìý(*Partially known, otherwise conjectured.) Ìý