Rigidity of the Torelli subgroup in Out(F N )

• Sebastian Hensel ^[3] ; Camille Horbez ^[1] ; Richard D. Wade ^[2]
  1. [1] University of Paris-Saclay

     University of Paris-Saclay

     Arrondissement de Palaiseau, Francia

     
  2. [2] University of Oxford

     University of Oxford

     Oxford District, Reino Unido

     
  3. [3] Universität München, Munich, Germany

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 41, Nº 1, 2025, págs. 73-112
• Idioma: inglés
• DOI: 10.4171/RMI/1505
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• Resumen

  • Let N≥4. We prove that every injective homomorphism from the Torelli subgroup IA N to Out(FN) differs from the inclusion by a conjugation in Out(FN). This applies more generally to the following subgroups of Out(FN): every finite-index subgroup of Out(F N) (recovering a theorem of Farb and Handel); every subgroup of Out(FN) that contains a finite-index subgroup of one of the groups in the Andreadakis–Johnson filtration of Out(FN); every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich subgroup of Out(FN) –those are subgroups that contain sufficiently many twists in an appropriate sense.

    Among applications, this recovers the fact that the abstract commensurator of every group above is equal to its relative commensurator in Out(FN); it also implies that all subgroups in the Andreadakis–Johnson filtration of Out(FN) are co-Hopfian.

    We also prove the same rigidity statement for subgroups of Out(F3) which contain a power of every Nielsen transformation. This shows, in particular, that Out(F3) and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the N=3 case.