Maps and Groups
by
Thomas W. Tucker
Colgate University
Coauthors: Bruce Richter, Jozef Širáň, Robert Jajcay, Mark Watkins

Oriented maps correspond to permutation groups (called, variously, dart or monodromy or cartographic groups) generated by two elements, one an involution. This observation is well-known and is exploited, for example, in the computation of all regular maps of small genus. Less recognized is the correspondence between branched coverings of maps and homomorphisms of permutation groups. This association is especially useful in understanding maps with extra symmetry: Cayley maps and edge-transitive maps, in addition to regular maps. Questions about these maps often lead to curious questions about permutation groups. For example, determining when the (d;p, q) tesselation of the hyperbolic plane is a Cayley map is equivalent to determining when a long cycle in the symmetric group on d symbols can be factored as the product of two elements, one of order p and the other of order q (a question considered by G.A. Miller 100 years ago). As another example, the existence of a certain type of edge-transitive map depends on the existence of a group having no automorphism permuting a certain type of generating set. This talk will consider a number of such questions. It is based on joint work with Bruce Richter, Jozef Siran, Robert Jajcay, and Mark Watkins.

Date received: October 18, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafp-04.