Regular coverings and parallel products of Farey maps

TL;DR

This paper explores the structure of Farey maps and their relations via regular coverings and parallel products, revealing their spectra and extending the analysis to maps associated with Hecke groups.

Contribution

It introduces a detailed analysis of Farey maps' structure and spectra, and extends the framework to maps defined by Hecke groups, highlighting new structural insights.

Findings

Complete spectra of Farey maps are determined.

Farey maps are related through regular coverings and parallel products.

Extension of analysis to maps from Hecke groups.

Abstract

We examine the structure of Farey maps, which are a class of maps (graph embeddings on surfaces) that have received significant attention recently. We describe how they are related to each other through regular coverings and parallel products, and use these observations to find their complete spectra, recovering some known results. We then examine a similar class of maps defined by Hecke groups.