On infinite regular and chiral maps

TL;DR

This paper proves that infinite regular and chiral maps are confined to surfaces with at most one end, specifically on the Loch Ness monster surface for orientable cases with genus, highlighting topological constraints.

Contribution

It establishes topological limitations of infinite regular and chiral maps, showing their realization only on the Loch Ness monster surface under certain conditions.

Findings

Infinite regular and chiral maps occur on surfaces with at most one end.

Such maps on orientable surfaces with genus are only realizable on the Loch Ness monster.

The results constrain the topological surfaces supporting these maps.

Abstract

We prove that infinite regular and chiral maps take place on surfaces with at most one end. Moreover, we prove that an infinite regular or chiral map on an orientable surface with genus can only be realized on the Loch Ness monster, that is, the topological surface of infinite genus with one end.