For every d≥2, the canonical word-length spectral triple of (Z/2Z)≀F_d is not a spectral metric space, giving the first family of such counterexamples.
Connes,Compact metric spaces, Fredholm modules, and hyperfiniteness,Ergodic Theory Dynam
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Word-Length Spectral Triples of $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$ Are Not Metric
For every d≥2, the canonical word-length spectral triple of (Z/2Z)≀F_d is not a spectral metric space, giving the first family of such counterexamples.