Out(Fn) and the spectral gap conjecture

For n > 2, given φ1, …,φn randomly chosen isometries of S², it is well known that the group Γ generated by φ1, …,φn acts ergodically on S². In 1999, Gamburd, Jakobson, and Sarnak conjectured that for almost every choice of φ1,…,φn, this action is strongly ergodic. This is equivalent to the spectrum of ${\boldsymbol\varphi}_\mathbf1\boldsymbol+\boldsymbol\varphi_\mathbf1^{\boldsymbol-\mathbf1}\boldsymbol+\boldsymbol\dots\boldsymbol+{\boldsymbol\varphi}_\mathbf n\boldsymbol+\boldsymbol\varphi_\mathbf n^{\boldsymbol-\mathbf1}$ as an operator on L²(S²) having a spectral gap, that is, all eigenvalues but the largest one being bounded above by some λ1 > 2n. (The largest eigenvalue λ0, corresponding to constant functions, is 2n.) In this paper, we show that if n > 2, then either the conjecture is true or almost every n-tuple fails to have a gap. In fact, the same result holds for any n-tuple φ1,…,φn in any compact group K that is an almost direct product of SU(2) factors with L²(S²) replaced by L²(x), where X is any homogeneous K space. A weaker result is proven for n = 2 and some conditional results for similar actions of Fn on homogeneous spaces for more general compact groups.