$\Out(F_n)$ and the spectral gap conjecture
Abstract
Description
For $n>2$, given $ϕ_1,...,ϕ_n$ randomly chosen isometries of $S^2$, it is well-known that the group $\G$ generated by $ϕ_1,...,ϕ_n$ acts ergodically on $S^2$. It is conjectured in \cite{GJS} that for almost every choice of $ϕ_1,...,ϕ_n$ this action is {\em strongly ergodic}. This is equivalent to the spectrum of $ϕ_1+ϕ_1{\inv}+{...}+ϕ_n+ϕ_n^{\inv}$ 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 $λ_1<2n$. (The largest eigenvalue $λ_0$, 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 $ϕ_1,..., ϕ_n$ 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.
Final version. Minor modifications to text, several references added. To appear IMRN
Final version. Minor modifications to text, several references added. To appear IMRN