Random length-spectrum rigidity for free groups

Abstract

We say that a subset S ⊆ F N S\subseteq F_N is spectrally rigid if whenever T 1 , T 2 ∈ c v N T_1, T_2\in \mathrm {cv}_N are points of the (unprojectivized) outer space such that | | g | | T 1 = | | g | | T 2 ||g||_{T_1}=||g||_{T_2} for every g ∈ S g\in S , then T 1 = T 2 T_1=T_2 in c v N \mathrm {cv}_N . It is well known that F N 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 F_N . We prove that if A A is a free basis of F N F_N (where N ≥ 2 N\ge 2 ), then almost every trajectory of a non-backtracking simple random walk on F N F_N with respect to A A is a spectrally rigid subset of F N F_N .