Word-length spectral triples as compact quantum metric spaces
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Statement
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 the canonical spectral triple of the Lamplighter group , with the word-length function of a finite symmetric generating set, fails to be a spectral metric space.
Context
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.
A recognised open question in non-commutative metric geometry, sitting in Rieffel's compact-quantum-metric-space programme, where the positive cases (polynomial growth, word-hyperbolic) were known and the expected negative answer had resisted an explicit witness. Specialist, with no Wikipedia article of its own, so scored with the resolved named problems of a single subfield rather than the broadly tracked conjectures.
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The acknowledgements state: "The author acknowledges the use of GPT-5.6 Sol as an exploratory tool to assist in finding the counterexample. The AI was used under the author's strict mathematical guidance. All mathematical content and arguments were rigorously reviewed, verified, and substantially revised by the author, who assumes full responsibility for the final manuscript." The model helped find the central object, which is an essential named step, but the framing is explicitly human-led and the author reports substantially revising everything - the co-developed tier rather than the discovered one.
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