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Asciak, Kirstie Alice
(2024).
DOI: https://doi.org/10.21954/ou.ro.00099822
Abstract
A map is a cellular embedding of a connected graph on a closed surface. An automorphism of a map is a permutation of its flags that preserves the cell structure of the map. The group of all automorphisms of a map acts freely on the flag set of the map. If this action is also transitive, and hence regular, the map itself is regular. Regularity of a map can be interpreted as the largest level of ‘internal symmetry’ a map exhibits. A regular map of type (k, m), that is of face length m and valency k, can be identified with smooth quotients of extended (2, k, m )-triangle groups, which are generated by three involutions whose products have prescribed orders 2, m and k. Similarly, orientably-regular maps of the same type can be identified with smooth quotients of the ordinary (2, k, m )-triangle groups, constituting the ‘even’ subgroup of index 2 of the extended (2,k, m )-triangle group. This Thesis considers construction of regular and orientably-regular maps with specified external symmetries, in particular, with specified invariance to rotational powers. Given a map M of type (k, m ) and an integer j relatively prime to k, the operator of a jth rotational power constructs a new map M(j) from M by replacing all the local rotations by their jth powers. If M is (orientably-) regular, then so is M(j). If M(j) is isomorphic to M, then j is an exponent of M. The collection of exponents of M forms a group isomorphic to a subgroup of the group of units mod k. We consider the following: 1. Given k and a group U of units mod k, does there exist an (orientably-)regular map of valency k with exponent group U? 2. Given m and k, does there exist an (orientably-) regular map of type (k, m) with trivial exponent group?
CORE (COnnecting REpositories)
CORE (COnnecting REpositories)
