Random length-spectrum rigidity for free groups

Abstract:

We say that a subset $S\subseteq F_N$ is spectrally rigid if whenever $T_1, T_2\in \mathrm {cv}_N$ 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 $\mathrm {cv}_N$. 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 $F_N$.