External Symmetries and Asymetries of Maps
by
M. Škoviera
Bratislava

Every map on an oriented surface can be described combinatorially as a pair (K, R), where K is a connected graph and R is a rotation of K, a permutation of the set of directed edges (arcs) of K which cyclically permutes arcs emanating from the same vertex. Within the framework of this combinatorial representation, symmetries of a map - map automorphisms - correspond to graph automorphisms which commute with the rotation: \phiR=R\phi. A different type of symmetries is obtained when reflections of a map are taken into account; in this case \phiR=R^-1\phi. From the point of view of oriented maps, a reflection cannot be counted as an automorphism for it does not preserve the given orientation of the surface. Nevertheless, it is an example of an external ``automorphism" - exomorphism - of the map. The general definition of an exomorphism of an oriented map leads to the concept of an exponent of a map. An exponent of a map is any integer e such that the map (K, R^e) is isomorphic to the original map (K, R). A similar concept can be introduced for unoriented maps, those which can be described by means of three involutions (rotary, longitudinal and transversal) acting on the flags of the map. In both cases, exponents reduced modulo the valency of the map form an Abelian group. This group can be viewed as a measure of additional, external symmetries of maps. In the talk we will survey the motivation, basic properties, and applications of exomorphisms and exponents of maps with the emphasis on the construction and classification of regular maps on surfaces. Attention will also be paid to mirror asymetrical regular maps - those that do not admit -1 as an exponent. These maps are commonly known as chiral maps and have been an object of a special interest because they are very rare. We show that mirror asymetry can be measured and that there exist extremely mirror asymetrical regular maps.

Date received: May 26, 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 # cafd-24.