The fractal nature of groups of automorphisms of K3 surfaces

The group of automorphisms of a K3 surface generates a tiling of a hyperbolic space of dimension n-1, where n is the Picard number for the K3 surface.  If the K3 surface contains a -2 curve, then the tiles have infinite volume.  When n = 4, the hyperbolic space can be modeled by the Poincaré upper half space, where hyperbolic planes are represented by half spheres (or half planes) perpendicular to the plane z = 0.   The plane z = 0 represents the boundary at infinity of the hyperbolic space.  The tiling of the space generates a tiling of the plane z = 0.  The light blue lines in this picture is the tiling of z = 0 induced by the group of automorphisms of a K3 surface with Picard number 4. The K3 surface is generated by a smooth (2,2,2) form in P1xP1xP1 that contains a single smooth line (which generates all of its -2 curves).  The dark blue represent planes in hyperbolic three space.  These planes form the boundary of a cross section of the Kahler cone (or nef cone).  There are two obvious points of interest in this picture -- the point where many small dark blue circles accumulate, and the point of tangency of many light blue circles.  In the figures below, these points are chosen to be the points at infinity.  For more information, see the preprint "The nef cone and dynamics on a K3 surface." (Caution:  Large file -- 10MB.)

-Arthur Baragar, November 5th, 2003, revised May 14th, 2004.

This material is based upon work supported by the National Science Foundation under Grant No. 0403686.  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundtaion.
Tiling at infinity.