Spectral Rigidity and Subgroups of Free Groups

Abstract

A subset $\Sigma \subset F_N$ of the free group of rank $N$ is called \emph{spectrally rigid} if whenever trees $T,T^{\prime }$ in Culler-Vogtmann Outer Space are such that $\Vert g{\Vert }_T=\Vert g{\Vert }_{T^{\prime }}$ for every $g\in \Sigma$, it follows that $T=T^{\prime }$. Results of Smillie, Vogtmann, Cohen, Lustig, and Steiner prove that (for $N\ge 2$) 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 $H⊲F_N$ is spectrally rigid.

arXiv comment10 pages