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arXiv · 2603.17869

On the spectral gap conjecture for pairs in SU(2)

Oleg Pikhurko·Kohki Sakamoto
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Abstract

For $n \ge 2$, Gamburd, Jakobson, and Sarnak [J. Eur. Math. Soc. 1, 51-85 (1999)] conjectured that almost every $n$-tuple in $\mathrm{SU}(2)$ has a spectral gap. Toward this conjecture, Fisher [Int. Math. Res. Not. (2006)] established a zero-one law for $n \ge 3$, but obtained only a partial result for $n=2$. In this paper, we prove that the zero-one law also holds for $n=2$. We also remark that a Baire categorical analogue of this result holds.

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Oleg Pikhurko, Kohki Sakamoto. 2026-03-18. On the spectral gap conjecture for pairs in SU(2). https://arxiv.org/abs/2603.17869

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