Maps admitting trialities but not dualities

TL;DR

This paper constructs infinite families of maps on surfaces that are symmetric under certain order-3 operations but not under order-2 duality operations, using group theory.

Contribution

It introduces a group-theoretic method to identify maps invariant under triality but not duality.

Findings

Constructed infinite families of such maps.

Demonstrated maps invariant under Wilson's order-3 operations.

Showed absence of invariance under duality and Petrie duality.

Abstract

We use group theory to construct infinite families of maps on surfaces which are invariant under Wilson's map operations of order 3 but not under the operations of order 2, such as duality and Petrie duality.