# The planar Cayley graphs are effectively enumerable II

## TL;DR

This paper characterizes groups with planar, finitely generated Cayley graphs through a new type of group presentation called planar presentation, enabling their effective enumeration and answering a longstanding question about planar groups.

## Contribution

It introduces planar presentations and proves their equivalence to groups with planar Cayley graphs, providing an algorithmic recognition method and effective enumeration.

## Key findings

- Planar presentations can be recognized algorithmically.
- Groups with planar Cayley graphs are effectively enumerable.
- Affirmative answer to the effective enumeration of planar groups.

## Abstract

We show that a group admits a planar, finitely generated Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a consequence, we obtain an effective enumeration of the planar Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al. asking whether the planar groups can be effectively enumerated.