Lamplighter Groups Give Counterexamples to Metric Word-Length Spectral Triples
Mario Klisse constructs a family of lamplighter groups whose word-length spectral triples are not spectral metric spaces for any finite symmetric generating set, providing explicit counterexamples to a longstanding expectation; he reports using GPT-5.6 Sol exploratorily and substantially revising and verifying the resulting mathematics.
In a manuscript dated August 12, 2026, Mario Klisse gives a family of counterexamples to the expectation that word-length spectral triples always define compact quantum metric spaces. For every , he proves that the canonical spectral triple of the lamplighter group , equipped with the word length from any finite symmetric generating set, is not a spectral metric space.
The proof constructs averaging operators supported on lamp spheres whose commutators with the word-length operator remain uniformly bounded, while the operators themselves remain uniformly separated. This prevents the associated Connes pseudometric from recovering the weak-star topology on the state space. Klisse states that GPT-5.6 Sol was used as an exploratory tool in finding the counterexample under his mathematical guidance, and that he rigorously reviewed, verified, and substantially revised all arguments.
The result supplies an explicit nonmetric family but leaves related questions open, including whether the metric property can depend on the generating set and whether analogous counterexamples exist among finitely generated amenable or rapid-decay groups.