Spectral triples for finitely generated groups, index 0

TL;DR

This paper constructs spectral triples for all finitely generated groups using Cayley graphs and Clifford algebras, classifying groups into three types based on the properties of the Dirac operator.

Contribution

It provides a uniform method to build spectral triples for finitely generated groups and introduces a new classification based on the Dirac operator's index.

Findings

Spectral triples with non-trivial phase are constructed for all finitely generated groups.

The Dirac operator $D_{+}$ has index 0 for these spectral triples.

A natural classification of finitely generated groups into three types is proposed.

Abstract

Using Cayley graphs and Clifford algebras, we are able to give, for every finitely generated groups, a uniform construction of spectral triples with a generically non-trivial phase for the Dirac operator. Unfortunatly $D_{+}$ is index $0$, but we are naturally led to an interesting classification of finitely generated groups into three types.