Word-length spectral triples as compact quantum metric spaces

A countable discrete group with a proper length function carries a natural spectral triple on its reduced group C*-algebra. A well-studied question in non-commutative metric geometry asks whether the associated Connes pseudo-metric always recovers the weak-* topology on the state space, making it a compact quantum metric space in Rieffel's sense. It holds for groups of polynomial growth and for word-hyperbolic groups, and it was widely expected that not every word-length function works - but no explicit counterexample was known. False: for every integer $d \ge 2$ the canonical spectral triple of the Lamplighter group $(\mathbb{Z}/2\mathbb{Z}) \wr \mathbb{F}_d$, with the word-length function of a finite symmetric generating set, fails to be a spectral metric space.

The first explicit counterexample rather than a first suspicion: the abstract is clear that the failure was widely expected and that what was missing was a witness. It gives an infinite family, one for each d >= 2, all Lamplighter groups over free groups.