Quantitative Hilbert–Schmidt stability of lamplighter relations

For approximate finite-dimensional unitary representations of the lamplighter group, seek nearby exact representations with polynomial control of Hilbert–Schmidt stability rate and radius. The hypotheses impose approximate order-two and orbit-commutator relations through 2M, with M polynomial in the target tolerance.

Why it matters and what counts as progress

Why it matters: It asks for explicit quantitative stability in an infinitely presented group, where an uncontrolled existence statement would give less information.

What counts as progress: A smaller polynomial dependence of stability radius or rate on tolerance, or an explicit nearby exact representation under weaker approximate relations.

How the problem developed

22 Jul 2026 · Quantitative stability result [1]
Alon Dogon and Thomas Vidick Established explicit polynomial bounds on the lamplighter group's Hilbert–Schmidt stability rate and radius, answering a question of Dogon, Levit and Vigdorovich.

AI contribution

Polynomial Hilbert-Schmidt stability of the lamplighter group

Alon Dogon, Thomas Vidick

For finite-dimensional unitaries satisfying approximate order-two and orbit-commutator relations through 2M, with M polynomial in the target tolerance, Dogon and Vidick prove nearby exact lamplighter relations. Their argument gives polynomial upper bounds on Hilbert–Schmidt stability rate and radius.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (Lean 4 with Mathlib and statement-only Comparator)

Human understandingFull