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arXiv2014

Spectral Rigidity and Subgroups of Free Groups

A subset $Σ\subset F_N$ 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 $g \in Σ$, 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.

math.GR↗

arXiv2011

Non-Rigidity of Cyclic Automorphic Orbits in Free Groups

We say a subset $Σ\subseteq F_N$ 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 $|σ|_{T_1} = |σ|_{T_2}$ for every $σ\in Σ$, 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 $Σ$ is spectrally rigid. Capitalizing on their constructions, we prove that for any $Φ\in \Aut(F_N)$ and $g \in F_N$, the set $Σ= {Φ^n(g)}_{n \in \mathbb{Z}}$ is not spectrally rigid.

math.GR↗