Spectral Rigidity and Subgroups of Free Groups

A subset $\Sigma \subset F_N$ of the free group of rank $N$ is called \emph{spectrally rigid} if whenever trees $T, T'$ in Culler-Vogtmann Outer Space are such that $\| g \|_T = \| g \|_{T'}$ for every $g \in \Sigma$, it follows that $T = T'$. Results of Smillie, Vogtmann, Cohen, Lustig, and Steiner prove that (for $N \geq 2$) no finite subset of $F_N$ is spectrally rigid in $F_N$. We prove that if $\{ H_i \}_{i=1}^k$ is a finite collection of subgroups, each of infinite index, and $g_i \in F_N$, then $\cup_{i=1}^k g_i H_i$ is not spectrally rigid in $F_N$. Taking $H_i = 1$, we recover the results about finite sets. We also prove that any coset of a nontrivial normal subgroup $H \lhd F_N$ is spectrally rigid.