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*

August, 1995