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

TL;DR

This paper proves that almost all pairs in SU(2) have a spectral gap, confirming a conjecture and extending previous results by establishing a zero-one law for the case n=2.

Contribution

It establishes a zero-one law for spectral gaps in pairs in SU(2), completing the proof for n=2 and supporting the conjecture for almost all n-tuples.

Findings

Zero-one law holds for n=2 in SU(2).

Spectral gap conjecture is supported for almost all pairs.

Baire category analogue also holds.

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.