Abelian regular coverings of the quaternion hypermap

TL;DR

This paper investigates abelian regular coverings of the quaternion hypermap, classifies certain types of coverings, and explores their symmetry properties using algebraic and combinatorial methods.

Contribution

It introduces normalized multicyclic coverings of regular hypermaps, shows their transformation groups are nilpotent, and classifies abelian bicyclic coverings with symmetry and smoothness conditions.

Findings

Covering transformation groups are nilpotent with bounded class.

Classification of abelian normalized bicyclic coverings over the quaternion hypermap.

Explicit determination of coverings with various symmetry and smoothness levels.

Abstract

A hypermap is an embedding of a connected hypergraph into an orientable closed surface. A covering between hypermaps is a homomorphism between the embedded hypergraphs which extends to an orientation-preserving covering of the supporting surfaces. A covering of a hypermap onto itself is an automorphism, and a hypermap is regular if its automorphism group acts transitively on the brins. Depending on the algebraic theory of regular hypermaps and hypermap operations, the abelian regular coverings over the quaternion hypermap are investigated. We define normalized multicyclic coverings between regular hypermaps, generalizing almost totally branched coverings studied in [K. Hu, R. Nedela, N.-E Wang, Branched cyclic regular coverings over platonic maps, European J. Combin. 36 (2014) 531--549]. It is shown that the covering transformation group of a normalized multicyclic covering is a nilpotent group with bounded class. As an application the abelian normalized bicyclic coverings over the quaternion hypemap are classified. In particular, those coverings which possess various level of external symmetry or fulfil certain smoothness conditions are explicitly determined.