$\Out(F_n)$ and the spectral gap conjecture
  For $n>2$, given $\phi_1,...,\phi_n$ randomly chosen isometries of $S^2$, it is well-known that the group $\G$ generated by $\phi_1,...,\phi_n$ acts ergodically on $S^2$. It is conjectured in \cite{GJS} that for almost every choice of $\phi_1,...,\phi_n$ this action is {\em strongly ergodic}. This is equivalent to the spectrum of $\phi_1+\phi_1{\inv}+{...}+\phi_n+\phi_n^{\inv}$ as an operator on $L^2(S^2)$ having a spectral gap, i.e. all eigenvalues but the largest one being bounded above by some $\lambda_1<2n$. (The largest eigenvalue $\lambda_0$, corresponding to constant functions, is $2n$.)   In this article 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 is holds for any $n$-tuple $\phi_1,..., \phi_n$ in any any compact group $K$ that is an almost direct product of SU(2) factors with $L^2(S^2)$ replaced by $L^2(X)$ where $X$ is any homogeneous $K$ space. A weaker result is proven for $n=2$ and some conditional results for similar actions of $F_n$ on homogeneous spaces for more general compact groups. 
David Fisher