Spectral triples for finitely generated groups, index 0

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.

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