Spectral rigidity of automorphic orbits in free groups

Details Spectral rigidity of automorphic orbits in free groups Spectral rigidity of automorphic orbits in free groups

2011-06-03

arxiv.org/abs/1106.0688v3

Mathematics

Group Theory

The proofs of spectral rigidity of the set of primitive elements and of the existence of finite relatively rigid sets are repl

Scientific paper

It is well-known that a point $T\in cv_N$ in the (unprojectivized) Culler-Vogtmann Outer space $cv_N$ is uniquely determined by its \emph{translation length function} $||.||_T:F_N\to\mathbb R$. A subset $S$ of a free group $F_N$ is called \emph{spectrally rigid} if, whenever $T,T'\in cv_N$ are such that $||g||_T=||g||_{T'}$ for every $g\in S$ then $T=T'$ in $cv_N$. By contrast to the similar questions for the Teichm\"uller space, it is known that for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$. In this paper we prove that for $N\ge 3$ if $H\le Aut(F_N)$ is a subgroup that projects to an infinite normal subgroup in $Out(F_N)$ then the $H$-orbit of an arbitrary nontrivial element $g\in F_N$ is spectrally rigid. We also establish a similar statement for $F_2=F(a,b)$, provided that $g\in F_2$ is not conjugate to a power of $[a,b]$.

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