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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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References:
[1] M Bestvina, M Feighn, Hyperbolicity of the complex of free factors · Zbl 1190.20017
[2] M Bestvina, M Feighn, Outer limits, preprint (1993)
[3] M Bestvina, M Feighn, A hyperbolic \(\mathrm{Out}(F_n)\)-complex, Groups Geom. Dyn. 4 (2010) 31 · Zbl 1190.20017
[4] M Bestvina, M Feighn, M Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997) 215 · Zbl 0884.57002
[5] M Bestvina, M Handel, Train tracks and automorphisms of free groups, Ann. of Math. 135 (1992) 1 · Zbl 0757.57004
[6] F Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. 124 (1986) 71 · Zbl 0671.57008
[7] F Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139 · Zbl 0653.32022
[8] M R Bridson, K Vogtmann, The symmetries of outer space, Duke Math. J. 106 (2001) 391 · Zbl 1037.20025
[9] M R Bridson, K Vogtmann, Automorphism groups of free groups, surface groups and free abelian groups, Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 301 · Zbl 1184.20034
[10] I Chiswell, Introduction to \(\Lambda\)-trees, World Scientific Publishing Co. (2001) · Zbl 1004.20014
[11] M Clay, A Pettet, Currents twisting and nonsingular matrices, Comment. Math. Helv. (to appear) · Zbl 1286.20049
[12] M M Cohen, M Lustig, Very small group actions on \(\mathbbR\)-trees and Dehn twist automorphisms, Topology 34 (1995) 575 · Zbl 0844.20018
[13] M M Cohen, M Lustig, M Steiner, \(\mathbbR\)-tree actions are not determined by the translation lengths of finitely many elements, Math. Sci. Res. Inst. Publ. 19, Springer (1991) 183 · Zbl 0826.20028
[14] M Cohen, W Metzler, A Zimmermann, What does a basis of \(F(a,\thinspace b)\) look like?, Math. Ann. 257 (1981) 435 · Zbl 0458.20028
[15] T Coulbois, A Hilion, M Lustig, \(\mathbbR\)-trees and laminations for free groups I: Algebraic laminations, J. Lond. Math. Soc. 78 (2008) 723 · Zbl 1197.20019
[16] T Coulbois, A Hilion, M Lustig, \(\mathbbR\)-trees and laminations for free groups II: The dual lamination of an \(\mathbbR\)-tree, J. Lond. Math. Soc. 78 (2008) 737 · Zbl 1198.20023
[17] C B Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990) 150 · Zbl 0704.53035
[18] C B Croke, Rigidity theorems in Riemannian geometry, IMA Vol. Math. Appl. 137, Springer (2004) 47
[19] C B Croke, P Eberlein, B Kleiner, Conjugacy and rigidity for nonpositively curved manifolds of higher rank, Topology 35 (1996) 273 · Zbl 0859.53024
[20] C Croke, A Fathi, J Feldman, The marked length-spectrum of a surface of nonpositive curvature, Topology 31 (1992) 847 · Zbl 0779.53025
[21] M Culler, Finite groups of outer automorphisms of a free group, Contemp. Math. 33, Amer. Math. Soc. (1984) 197 · Zbl 0552.20024
[22] M Culler, J W Morgan, Group actions on \(\mathbbR\)-trees, Proc. London Math. Soc. 55 (1987) 571 · Zbl 0658.20021
[23] M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91 · Zbl 0589.20022
[24] M Culler, K Vogtmann, The boundary of outer space in rank two, Math. Sci. Res. Inst. Publ. 19, Springer (1991) 189 · Zbl 0786.57002
[25] F Dal’Bo, I Kim, Marked length rigidity for symmetric spaces, Comment. Math. Helv. 77 (2002) 399 · Zbl 1002.22005
[26] M Duchin, C J Leininger, K Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010) 231 · Zbl 1207.53052
[27] A Fathi, F Laudenbach, V Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France (1979) 284
[28] S Francaviglia, Geodesic currents and length compactness for automorphisms of free groups, Trans. Amer. Math. Soc. 361 (2009) 161 · Zbl 1166.20032
[29] S Francaviglia, A Martino, Metric properties of outer space, Publ. Mat. 55 (2011) 433 · Zbl 1268.20042
[30] V Guirardel, Approximations of stable actions on \(\mathbbR\)-trees, Comment. Math. Helv. 73 (1998) 89 · Zbl 0979.20026
[31] V Guirardel, Dynamics of \(\mathrm{Out}(F_n)\) on the boundary of outer space, Ann. Sci. École Norm. Sup. 33 (2000) 433 · Zbl 1045.20034
[32] U Hamenstädt, Invariant Radon measures on measured lamination space, Invent. Math. 176 (2009) 223 · Zbl 1209.37023
[33] M Handel, L Mosher, Subgroup classification in \(\mathrm{Out}(F_n)\) · Zbl 1285.20033
[34] S Hersonsky, F Paulin, On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv. 72 (1997) 349 · Zbl 0908.57009
[35] I Kapovich, The frequency space of a free group, Internat. J. Algebra Comput. 15 (2005) 939 · Zbl 1110.20031
[36] I Kapovich, Currents on free groups (editors R Grigorchuk, M Mihalik, M Sapir, Z Sunik), Contemp. Math. 394, Amer. Math. Soc. (2006) 149 · Zbl 1110.20034
[37] I Kapovich, Clusters, currents, and Whitehead’s algorithm, Experiment. Math. 16 (2007) 67 · Zbl 1158.20014
[38] I Kapovich, Random length-spectrum rigidity for free groups, Proc. Amer. Math. Soc. 140 (2012) 1549 · Zbl 1268.20043
[39] I Kapovich, M Lustig, The actions of \(\mathrm{Out}(F_k)\) on the boundary of outer space and on the space of currents: minimal sets and equivariant incompatibility, Ergodic Theory Dynam. Systems 27 (2007) 827 · Zbl 1127.20025
[40] I Kapovich, M Lustig, Geometric intersection number and analogues of the curve complex for free groups, Geom. Topol. 13 (2009) 1805 · Zbl 1194.20046
[41] I Kapovich, M Lustig, Domains of proper discontinuity on the boundary of outer space, Illinois J. Math. 54 (2010) 89 · Zbl 1259.20050
[42] I Kapovich, M Lustig, Intersection form, laminations and currents on free groups, Geom. Funct. Anal. 19 (2010) 1426 · Zbl 1242.20052
[43] I Kapovich, M Lustig, Stabilizers of \(\mathbbR\)-trees with free isometric actions of \(F_N\), J. Group Theory 14 (2011) 673 · Zbl 1262.20031
[44] I Kapovich, T Nagnibeda, The Patterson-Sullivan embedding and minimal volume entropy for outer space, Geom. Funct. Anal. 17 (2007) 1201 · Zbl 1135.20031
[45] I Kapovich, T Nagnibeda, Geometric entropy of geodesic currents on free groups, Contemp. Math. 532, Amer. Math. Soc. (2010) 149 · Zbl 1216.20034
[46] I Kapovich, T Nagnibeda, Generalized geodesic currents on free groups · Zbl 1216.20034
[47] I Kim, Ergodic theory and rigidity on the symmetric space of non-compact type, Ergodic Theory Dynam. Systems 21 (2001) 93 · Zbl 0978.37016
[48] I Kim, Marked length rigidity of rank one symmetric spaces and their product, Topology 40 (2001) 1295 · Zbl 0997.53034
[49] I Kim, Rigidity on symmetric spaces, Topology 43 (2004) 393 · Zbl 1049.53031
[50] R Martin, Non-uniquely ergodic foliations of thin-type, measured currents and automorphisms of free groups, PhD thesis, University of California, Los Angeles (1995)
[51] J P Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. 131 (1990) 151 · Zbl 0699.58018
[52] F Paulin, The Gromov topology on \(\mathbbR\)-trees, Topology Appl. 32 (1989) 197 · Zbl 0675.20033
[53] B Ray, Non-rigidity of cyclic automorphic orbits in free groups, Int. J. Alg. Comp. 22 (2012) 1250021 · Zbl 1264.20037
[54] J Smillie, K Vogtmann, Length functions and outer space, Michigan Math. J. 39 (1992) 485 · Zbl 0773.05058
[55] K Vogtmann, What is\(\dots\)outer space?, Notices Amer. Math. Soc. 55 (2008) 784 · Zbl 1194.20038
[56] B Zimmermann, Über Homöomorphismen \(n\)-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen, Comment. Math. Helv. 56 (1981) 474 · Zbl 0475.57015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.