Countable representability of Lie groups
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Does every second-countable Lie group admit a faithful action on a countable set, as an abstract group? No continuity of that permutation action is required; the related target extends to locally compact second-countable groups.
Why it matters and what counts as progress
Why it matters: Countable abstract permutation actions ask for algebraic representability without requiring a continuous action of the original topology.
What counts as progress: Construct a faithful countable action for the full class or prove a genuine obstruction in a broader class of groups.
How the problem developed
- 2021 · Partial result [1]
Nicolas Monod Established countable representability for nilpotent Lie groups, the first general non-linear family treated in this form.
- 2023 · Partial result [2]
Annalisa Conversano and Nicolas Monod Proved representability for solvable Lie groups and groups with a linear Levi component, reducing the remaining question to the simple case.
- 13 Oct 2026 · Reported theorem [3]
Nicolas Monod and Andreas Thom Reported that every Lie group acts faithfully by permutations on a countable set, answering Ulam’s question.
AI contribution
Lie groups are countably representable
The model-origin deformation/Puiseux construction with prescribed residues used in the countable-representability argument, separating the authors’ general Lie-group extensions.
Sources
Website timeline
Publication dates and editorial changes.