Simplicity of C*-algebras associated to higher-rank graphs

We prove that if \Lambda is a row-finite k-graph with no sources, then the associated C*-algebra is simple if and only if \Lambda is cofinal and satisfies Kumjian and Pask's aperiodicity condition, known as Condition (A). We prove that the aperiodicity condition is equivalent to a suitably modified version of Robertson and Steger's original nonperiodicity condition (H3) which in particular involves only finite paths. We also characterise both cofinality and aperiodicity of \Lambda in terms of ideals in C*(\Lambda).