Accessibility in transitive graphs

TL;DR

This paper proves that for transitive graphs, the properties of their cycle and cut spaces are linked, confirming conjectures about their accessibility and providing a new proof for a fundamental group theory result.

Contribution

It establishes that the finiteness of the cycle space's generation implies the same for the cut space in transitive graphs, confirming key accessibility conjectures.

Findings

Cycle space being generated by bounded cycles implies cut space is finitely generated.

Locally finite hyperbolic transitive graphs are accessible.

Provides a combinatorial proof of Dunwoody's accessibility theorem.

Abstract

We prove that the cut space of any transitive graph $G$ is a finitely generated ${\rm Aut}(G)$-module if the same is true for its cycle space. This confirms a conjecture of Diestel which says that every locally finite transitive graph whose cycle space is generated by cycles of bounded length is accessible. In addition, it implies Dunwoody's conjecture that locally finite hyperbolic transitive graphs are accessible. As a further application, we obtain a combinatorial proof of Dunwoody's accessibility theorem of finitely presented groups.