Spectral triples and wavelets for higher-rank graphs

Abstract

In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph Λ, via the infinite path space ${\mathrm{\Lambda }}^{\infty }$ of Λ. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of ${\mathrm{\Lambda }}^{\infty }$ 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 Λ. 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 ζ-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure μ, and show that μ is a rescaled version of the measure M on ${\mathrm{\Lambda }}^{\infty }$ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of $L^2({\mathrm{\Lambda }}^{\infty },M)$ which was constructed by Farsi et al.