Automorphisms Adjacent to the Identity

If T is a triangulation of a closed 2-manifold, and f is an automorphism of T, then we say that f is adjacent to the identity if v is adjacent to f(v) for every vertex v of T.

Problem: Find all triangulations that have an automorphism adjacent to the identity.

There are exactly two such triangulations of the plane: the 6-regular triangulation and one coming from the octahedron. Another example is formed from C_3 x C_3 embedded in the torus by adding in each face the diagonal joining (i,j) to (i+1,j+1). This example is a natural quotient of the 6-regular triangulation of the plane. Other quotients of the 6-regular triangulation also admit automorphisms adjacent to the identity. The complete graph K_4 in the projective plane is yet another such example. It is a quotient of the octahedron.

If all orbits of points have length at least 3, then the examples in the previous paragraph are all such triangulations. In particular, the triangulation is a quotient of a planar example. The difficulty arises when there are orbits of length 2.

Submitted by: Steve Fisk, Department of Mathematics, Bowdoin College, Brunswick ME 04011-2599 USA.

Send comments to dan.archdeacon@uvm.edu and to fisk@bowdin.edu