# Mark Watkins

# Edge Transitive Maps on Orientable Surfaces

## Abstract for the Colloquium

Thursday, April 15, 1999

Jack Graver and I (1997) established that the automorphism group of an
edge-transitive, locally finite map manifests one of exactly 14
algebraically consistent combinations (called *types*) of the kinds of
stabilizers of its edges, its vertices, its faces, and its Petrie walks.
Exactly eight of these types are realized by infinite, locally finite maps
in the Euclidean or hyperbolic plane. H.S.M. Coxeter had previously
observed that the nine finite edge-transitive planar graphs realize three
of these eight planar types in the sphere.

In the past year, J. Siran, T. Tucker, and I have shown that for
each of the 14 types and each integer n >= 11 such that n is congruent to 3 or 11 (mod 12), there exist finite, orientable, edge-transitive
maps whose various stabilizers conform to the given type and whose
automorphism groups are (abstractly) isomorphic to the symmetric group
Sym(n). Exactly seven of these types (not a subset of the planar eight) are shown
to admit infinite families of finite, edge-transitive maps on the torus,
and their automorphism groups are determined explicitly. Thus all finite,
edge-transitive toroidal maps are classified according to this schema.