The planar Cayley graphs are effectively enumerable I: consistently planar graphs

TL;DR

This paper introduces a method to effectively enumerate finitely generated groups acting on 2-manifolds and their planar Cayley graphs, providing a systematic classification and answering a longstanding question.

Contribution

It presents a new type of group presentation that captures groups with certain actions and extends this to enumerate planar Cayley graphs effectively.

Findings

Effective enumeration of groups acting on 2-manifolds.

Effective enumeration of planar finitely generated Cayley graphs.

Affirmative answer to Droms et al.'s question.

Abstract

We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation capturing exactly these groups. Extending this in a companion paper, we find group presentations capturing the planar finitely generated Cayley graphs. Thus we obtain an effective enumeration of these Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al.