The spectral gap conjecture for random isometries of the sphere

Let $n\geq 2$, let $\phi_1,\ldots,\phi_n$ be a collection of elements of $SU(2)$, and let $L^2_0(SU(2))$ denote the orthogonal complement of the constant functions in $L^2(SU(2))$. The self-adjoint operator

$$\phi_1+\phi_1^{-1}+\cdots+\phi_n+\phi_n^{-1}$$

on $L^2_0(SU(2))$ has supremum of eigenvalues $\lambda_1\leq 2n$. The spectral gap conjecture. For almost every collection $\phi_1,\ldots,\phi_n$, one has $\lambda_1<2n$.

This is equivalent, in the sphere action formulation, to strong ergodicity for almost every randomly chosen collection of isometries. The paper studies the dichotomy between this conjecture and almost-everywhere failure of a spectral gap, but the conjecture is not resolved here.