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.
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