Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Aperiodicity and cofinality for finitely aligned higher-rank graphs


a1 School of Mathematics and Applied Statistics, Austin Keane Building (15), University of Wollongong, Wollongong, NSW, Australia. e-mail:


We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs Λ, and prove that C*(Λ) is simple if and only if Λ is aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms of finite paths. To prove our main result, we first characterise each of aperiodicity and cofinality of Λ in terms of the ideal structure of C*(Λ). In an appendix we show how our new cofinality condition simplifies in a number of special cases which have been treated previously in the literature; even in these settings our results are new.

(Received May 06 2009)

(Revised November 18 2009)

(Online publication May 10 2010)


† This research was supported by the Australian Research Council.