Accessibility in transitive graphs

Matthias Hamann

Published 2014-04-30, updated 2014-11-25Version 3

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.