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Spectral rigidity of automorphic orbits in free groups. (English) Zbl 1261.20040
Algebr. Geom. Topol. 12, No. 3, 1457-1486 (2012); corrigendum 14, No. 5, 3081-3088 (2014).
Let $$F_N$$ be the finitely generated free group of rank $$N\geq 2$$. The non-projectivized Outer space $$\mathrm{cv}_N$$ is introduced by M. Culler and K. Vogtmann [in Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)]. A basic fact in the theory of Outer space states that every $$T\in\mathrm{cv}_N$$ is uniquely determined by its translation length function $$\|\cdot\|\colon F_N\to\mathbb R$$, where for every $$g\in F_N$$ $$\|g\|_T=\min_{x\in T}d_T(x,gx)$$ is the translation length of $$g$$ (namely the Marked Length Rigidity Conjecture holds for Outer space).
A subset $$R\subseteq F_N$$ is called spectrally rigid if whenever $$T_1,T_2\in\mathrm{cv}_N$$ are such that $$\|g\|_{T_1}=\|g\|_{T_2}$$ for every $$g\in R$$, then $$T_1=T_2$$ in $$\mathrm{cv}_N$$. As noted above $$R=F_N$$ is spectrally rigid.
It is proved by J. Smillie and K. Vogtmann [Mich. Math. J. 39, No. 3, 485-493 (1992; Zbl 0773.05058)] for $$N\geq 3$$ and by M. M. Cohen, M. Lustig and M. Steiner [Publ., Math. Sci. Res. Inst. 19, 183-187 (1991; Zbl 0826.20028)] for $$N=2$$, that there does not exist a finite spectrally rigid subset of $$F_N$$. I. Kapovich [Proc. Am. Math. Soc. 140, No. 5, 1549-1560 (2012; Zbl 1268.20043)] has given a class of examples of spectrally rigid subsets of a free group.
In paper under consideration the authors obtain a very different class of examples of spectrally rigid subsets of free groups.
Their main result is: Theorem A. Let $$N\geq 2$$ and let $$H\leq\operatorname{Aut}(F_N)$$ be an ample subgroup. Let $$g\in F_N$$ be an arbitrary nontrivial element; in the case $$N=2$$ we also assume that $$g\in F_2=F(a,b)$$ is not conjugate to a nonzero power of $$[a,b]$$ in $$F_2$$. Then the orbit $$Hg=\{\varphi(g):\varphi\in H\}$$ is a spectrally rigid subset of $$F_N$$.
Here the subgroup $$H$$ is ample means that the image of $$H$$ in $$\mathrm{Out}(F_N)$$ contains an infinite normal subgroup of $$\mathrm{Out}(F_N)$$.
For the proof of this theorem the authors firstly prove that the set $$\mathcal P_N$$ of all the primitive elements in $$F_N$$ is a spectrally rigid subset in $$F_N$$ (Theorem 3.4 in the paper).
After that the authors use heavily the machinery of geodesic currents on free groups, and particularly exploit the geometric intersection form between trees and currents, constructed by I. Kapovich [Contemp. Math. 394, 149-176 (2006; Zbl 1110.20034)] and I. Kapovich and M. Lustig [Geom. Topol. 13, No. 3, 1805-1833 (2009; Zbl 1194.20046)].
The arguments for the proof of Theorem 3.4 are derived from S. Francaviglia and A. Martino [Publ. Mat., Barc. 55, No. 2, 443-473 (2011; Zbl 1268.20042)], these arguments give the following ‘relative rigidity’ result:
Theorem B. Let $$T\in\mathrm{cv}_N$$ be arbitrary. There exists a finite set $$S$$ (depending on $$T$$) of primitive elements in $$F_N$$ with the following property: Whenever $$T'\in\mathrm{cv}_N$$ is such that $$\|g\|_{T'}=\|g\|_T$$ for every $$g\in S$$ then $$T=T'$$ in $$\mathrm{cv}_N$$.
Theorem A applies to the cases where $$H=\operatorname{Aut}(F_N)$$ ($$N\geq 2$$) or where $$H\leq\operatorname{Aut}(F_N)$$ ($$N\geq 3$$) is the kernel of the natural homomorphism from $$\operatorname{Aut}(F_N)$$ to $$\operatorname{Aut}(F_N/\gamma_2(F_N))$$. This theorem also implies that for $$N\geq 3$$ any $$\operatorname{Aut}(F_N)$$-invariant subset of $$F_N$$ with more than one element is spectrally rigid in $$F_N$$.
The paper concludes with the discussion of several open problems motivated by the results of this paper.

##### MSC:
 20F65 Geometric group theory 20E05 Free nonabelian groups 20E36 Automorphisms of infinite groups 20F05 Generators, relations, and presentations of groups 20E08 Groups acting on trees 57M07 Topological methods in group theory 57M50 General geometric structures on low-dimensional manifolds 53C24 Rigidity results
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