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Corrigendum: “Spectral rigidity of automorphic orbits in free groups”. (English) Zbl 1307.20037
Summary: Lemma 5.1 in our paper mentioned in the title [Algebr. Geom. Topol. 12, No. 3, 1457-1486 (2012; Zbl 1261.20040)] says that every infinite normal subgroup of $$\mathrm{Out}(F_N)$$ contains a fully irreducible element; this lemma was substantively used in the proof of the main result, Theorem A [in loc. cit.]. Our proof of Lemma 5.1 [in loc. cit.] relied on a subgroup classification result of M. Handel and L. Mosher [“Subgroup classification in $$\mathrm{Out}(F_n)$$, arXiv:0908.1255], originally stated in [Handel and Mosher, loc. cit.] for arbitrary subgroups $$H\leq\mathrm{Out}(F_N)$$. It subsequently turned out [see M. Handel and L. Mosher, “Subgroup decomposition in $$\mathrm{Out}(F_n)$$: introduction and research announcement”, arXiv:1302.2681, page 1] that the proof of the Handel-Mosher theorem needs the assumption that $$H$$ is finitely generated. Here we provide an alternative proof of Lemma 5.1 from [Carette et al., loc. cit.], which uses the corrected version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of $$\mathrm{Out}(F_N)$$ on the free factor complex (due to Bestvina, Mann and Reynolds).

##### 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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