Vertex-transitive maps with Schl\"afli type {3, 7}

TL;DR

This paper classifies and constructs all vertex-transitive maps with Schl"afli type {3, 7} on certain surfaces, revealing their symmetry properties and providing a method to identify non-regular, non-chiral cases.

Contribution

It introduces a procedure to transform and classify all vertex-transitive maps with Schl"afli type {3, 7}, including non-regular, non-chiral maps on specific surfaces.

Findings

All such maps on surfaces with Euler characteristic between -1 and -40 are determined.

A method to convert these maps into 1- or 2-orbit maps is provided.

The automorphism groups of these maps have maximal possible order among equivelar vertex-transitive maps.

Abstract

Among all equivelar vertex-transitive maps on a given closed surface S, the automorphism groups of maps with Schl\"afli types {3, 7} and {7, 3} allow the highest possible order. We describe a procedure to transform all such maps into 1- or 2-orbit maps, whose symmetry type has been previously studied. In so doing we provide a procedure to determine all vertex-transitive maps with Schl\"afli type {3, 7} which are neither regular or chiral. We determine all such maps on surfaces with Euler characteristic -1 \geq \c{hi} \geq -40.