Algorithmic detectability of iwip automorphisms

We produce an algorithm that, given $\phi\in Out(F_N)$, where $N\ge 2$, decides wether or not $\phi$ is an iwip ("fully irreducible") automorphism.Comment: final version, to appear in the Bulletin of London Math. So

Random length-spectrum rigidity for free groups

We say that a subset $S\subseteq F_N$ is \emph{spectrally rigid} if whenever $T_1, T_2\in cv_N$ are points of the (unprojectivized) Outer space such that $||g||_{T_1}=||g||_{T_2}$ for every $g\in S$ then $T_1=T_2$ in \cvn. It is well-known that $F_N$ itself is spectrally rigid; it also follows from the result of Smillie and Vogtmann that there does not exist a finite spectrally rigid subset of $F_N$. We prove that if $A$ is a free basis of $F_N$ (where $N\ge 2$) then almost every trajectory of a non-backtracking simple random walk on $F_N$ with respect to $A$ is a spectrally rigid subset of $F_N$.Comment: 12 pages, no figures; to appear in Proceedings of the American Mathematical Society; updated ref to the Duchin-Leininger-Rafi pape

The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups

We show that if $H$ is a quasiconvex subgroup of infinite index in a non-elementary hyperbolic group $G$ then the Schreier coset graph $X$ for $G$ relative to $H$ is non-amenable (that is, $X$ has positive Cheeger constant). We present some corollaries regading the Martin boundary and Martin compactification of $X$ and the co-growth of $H$ in $G$.Comment: updated versio