Amenability versus strong Ulam stability of groups
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For countable discrete groups, is amenability equivalent to strong Ulam stability? Strong stability requires sufficiently accurate unitary almost representations on any complex Hilbert space to be uniformly close in operator norm to genuine representations on that same space.
Why it matters and what counts as progress
Why it matters: The converse would characterize amenability by robustness of approximate unitary representations, beyond groups containing nonabelian free subgroups.
What counts as progress: Prove the converse to the known amenable direction for new group classes or all countable groups, or construct a nonamenable strongly stable group.
How the problem developed
- 1982 · Partial result [1]
David Kazhdan Proved that amenable groups are strongly Ulam stable in operator norm, including representations on infinite-dimensional Hilbert spaces.
- 2013 · Open problem [1]
Marc Burger, Narutaka Ozawa and Andreas Thom Established failure of strong Ulam stability for groups containing nonabelian free subgroups and asked whether strong stability characterizes amenability.
- 5 Oct 2026 · Reported claim [2]
OpenAI The preprint claims that a countable discrete group is strongly Ulam stable exactly when it is amenable, using the operator norm and arbitrary Hilbert spaces.
AI contribution
Strong Ulam Stability Characterizes Amenability
For every countable discrete group, the manuscript claims amenability is equivalent to strong Ulam stability: sufficiently accurate unitary almost representations on any complex Hilbert space are uniformly close in operator norm to genuine representations on the same space. It gives a fixed-distance obstruction for nonamenable groups.
Sources
Website timeline
Publication dates and editorial changes.
OpenAI reports that strong Ulam stability characterizes amenability