Spectral triples and wavelets for higher-rank graphs

In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph Lambda, via the infinite path space Lambda(infinity) of Lambda. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of Lambda(infinity) which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary k-Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph A. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are zeta-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure mu and show that mu is a resealed version of the measure M on Lambda(infinity) which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrarni operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of L-2(Lambda(infinity), M) which was constructed by Farsi et al. (C) 2019 Elsevier Inc. All rights reserved.