Random length-spectrum rigidity for free groups

TL;DR

This paper proves that almost every trajectory of a non-backtracking simple random walk on a free group with respect to a basis is spectrally rigid, meaning it uniquely determines points in Outer space.

Contribution

It establishes that almost all random walk trajectories on free groups are spectrally rigid subsets, extending understanding of rigidity in free group Outer space.

Findings

Almost every random walk trajectory is spectrally rigid.

Finite subsets are not spectrally rigid.

Spectral rigidity characterizes points in Outer space.

Abstract

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 $F_N$.