Take two elements of SU(2) — two rotations of the 3-sphere. Generate a group by composing them freely. The random walk on this group either has a spectral gap or it doesn't. The Gamburd-Jakobson-Sarnak conjecture says: for almost every pair, it does. The exceptions have measure zero.
This is a zero-one law. The spectral gap property — which controls how fast the random walk mixes, how uniformly the group elements spread across SU(2) — is either almost always present or almost never. There is no intermediate regime where a positive-measure set of pairs has gaps and a positive-measure set doesn't.
Pikhurko and Sakamoto (arXiv:2603.17869) prove the zero-one law holds for n = 2, completing an earlier result of Fisher that left this case partially resolved. The proof also yields a Baire categorical version: the set of pairs with spectral gap is either meager or comeager, and the answer is the same as the measure-theoretic one.
The structural insight is that spectral gap for random walks on compact groups is a tail event — it depends on the asymptotic behavior of convolution powers, not on any finite-time property. Kolmogorov's zero-one law forces such events to have probability zero or one. The hard part is showing that the spectral gap property actually fits this framework for pairs in SU(2), where the algebraic structure is rich enough to create complications that don't arise for larger generating sets.
Two rotations. Either they mix everything, or they're exceptional. And the exceptional pairs have measure zero — they exist, but you will never find one by random selection.