Conjugacy classes of reflections of maps

TL;DR

This paper investigates the possible conjugacy classes of reflections in maps under different symmetry conditions, establishing bounds and constructing examples using topology and group theory.

Contribution

It provides new bounds on the number of conjugacy classes of reflections in edge-transitive maps and constructs explicit examples demonstrating various distributions.

Findings

Vertex- and face-transitive maps have no restrictions on reflection classes.

Edge-transitive maps have at most four classes of reflections.

Examples show diverse distributions of reflection classes are achievable.

Abstract

This paper considers how many conjugacy classes of reflections a map can have, under various transitivity conditions. It is shown that for vertex- and for face-transitive maps there is no restriction on their number or size, whereas edge-transitive maps can have at most four classes of reflections. Examples are constructed, using topology, covering spaces and group theory, to show that various distributions of reflections can be achieved. Connections with real forms of algebraic curves are also discussed.