Abstract:
This thesis concerns the theory of regular maps and the actions of the Wilson operators, determined by duality and Petrie duality, on them. The actions form four cases, and we will prove that in each case an instate family exists, and we shall provide an example of such a family. The most interesting of these cases occurs when a map admits neither duality or Petrie duality yet admits their product. This invariance is known as the 'triality' condition. It was not known whether an infinite family existed for this case until Jones and Poulton proved the existence of one in 2010. We will discuss this family, and use computational methods to find another.