Spectral triples and wavelets for higher-rank graphs

TL;DR

This paper constructs spectral triples for higher-rank graphs using their infinite path spaces, linking them to wavelet decompositions and analyzing associated spectral properties and eigenspaces.

Contribution

It introduces a new method to associate finitely summable spectral triples to higher-rank graphs via their infinite path spaces, connecting spectral geometry with wavelet analysis.

Findings

Spectral triples are $ta$-regular with explicit Dixmier trace formulas.

Eigenfunctions of Laplace-Beltrami operators refine wavelet decompositions.

The measure $mbda$ is a rescaled version of the measure $M$ on the infinite path space.

Abstract

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 $\Lambda^\infty$ of $\Lambda$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of $\Lambda^\infty$ 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 $\Lambda$. 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 rescaled version of the measure $M$ on $\Lambda^\infty$ 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 $L^2(\Lambda^\infty, M)$ which was constructed by Farsi et al.