Wavelets and spectral triples for higher-rank graphs

TL;DR

This paper introduces new spectral triples for higher-rank graphs, linking them to wavelet decompositions and measure theory on infinite path spaces, advancing the understanding of noncommutative geometric structures in graph algebras.

Contribution

It develops two novel methods to associate spectral triples with higher-rank graphs and connects these to wavelet decompositions and measure equivalences on infinite path spaces.

Findings

Spectral triples have regular zeta-functions with abscissa matching Hausdorff dimension.

The measure induced by the spectral triple coincides with a scaled Hausdorff measure.

Wavelet decompositions refine eigenspace decompositions of associated Laplace-Beltrami operators.

Abstract

In this paper, we present two new ways to associate a spectral triple to a higher-rank graph $\Lambda$. Moreover, we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of $\Lambda$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, to associate a family of ultrametric Cantor sets to a finite, strongly connected higher-rank graph $\Lambda$. Then we show that under mild hypotheses, the Pearson-Bellissard spectral triples of such Cantor sets have a regular $\zeta$-function, whose abscissa of convergence agrees with the Hausdorff dimension of the Cantor set, and that the measure $\mu$ induced by the associated Dixmier trace agrees with the measure $M$ on the infinite path space $\Lambda^\infty$ of $\Lambda$ which was introduced by an Huef, Laca, Raeburn, and Sims. Furthermore, we prove that $\mu = M$ is a rescaled version of the Hausdorff measure of the ultrametric Cantor set. From work of Julien and Savinien, we know that for $\zeta$-regular Pearson-Bellissard spectral triples, the eigenspaces of the associated Laplace-Beltrami operator constitute an orthogonal decomposition of $L^2(\Lambda^\infty, \mu)$; we show that this orthogonal decomposition refines the wavelet decomposition of Farsi et al. In addition, we generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras, and prove that the wavelet decomposition of Farsi et al.~describes the eigenspaces of its Dirac operator.