Planar transitive graphs

TL;DR

This paper proves that the first homology group of planar locally transitive graphs is finitely generated as an automorphism module, and establishes related results for fundamental groups of planar Cayley graphs, impacting group theory and graph accessibility.

Contribution

It introduces new finiteness results for homology and fundamental groups of planar graphs, extending understanding of their algebraic and topological properties.

Findings

First homology groups are finitely generated as automorphism modules.

Planar groups are finitely presented.

Planar locally finite transitive graphs are accessible.

Abstract

We prove that the first homology group of every planar locally transitive finite graph $G$ is a finitely generated ${\rm Aut}(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible.