Non-Rigidity of Cyclic Automorphic Orbits in Free Groups

TL;DR

This paper demonstrates that certain infinite automorphic orbits in free groups are not spectrally rigid, extending previous results and showing non-rigidity for orbits generated by automorphisms.

Contribution

It proves that automorphic orbits of the form \\{\\\Phi^n(g)\\\} are not spectrally rigid, generalizing earlier non-rigidity results to these infinite sets.

Findings

Automorphic orbits are not spectrally rigid.

Finite sets are not spectrally rigid, confirming previous results.

Infinite automorphic orbits lack spectral rigidity.

Abstract

We say a subset $\Sigma \subseteq F_N$ of the free group of rank $N$ is \emph{spectrally rigid} if whenever $T_1, T_2 \in \cv_N$ are $\mathbb{R}$-trees in (unprojectivized) outer space for which $|\sigma|_{T_1} = |\sigma|_{T_2}$ for every $\sigma \in \Sigma$, then $T_1 = T_2$ in $\cv_N$. The general theory of (non-abelian) actions of groups on $\mathbb{R}$-trees establishes that $T \in \cv_N$ is uniquely determined by its translation length function $|\cdot|_T \colon F_N \to \mathbb{R}$, and consequently that $F_N$ itself is spectrally rigid. Results of Smillie and Vogtmann \cite{MR1182503}, and of Cohen, Lustig, and Steiner \cite{MR1105334} establish that no finite $\Sigma$ is spectrally rigid. Capitalizing on their constructions, we prove that for any $\Phi \in \Aut(F_N)$ and $g \in F_N$, the set $\Sigma = {\Phi^n(g)}_{n \in \mathbb{Z}}$ is not spectrally rigid.