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Commensurations of Aut(FN) and Its Torelli Subgroup

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Abstract

For N≥3, the abstract commensurators of both Aut(FN) and its Torelli subgroup IAN are isomorphic to Aut(FN) itself.

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References

  1. Andreadakis, S.: On the automorphisms of free groups and free nilpotent groups. Proc. Lond. Math. Soc. 3(15), 239–268 (1965)

    Article  MathSciNet  Google Scholar 

  2. Bestvina, M., Bridson, M.R.: Rigidity of the free factor complex (2023). arXiv:2306.05941

  3. Bestvina, M.: The topology of Out(Fn). In: Proceedings of the International Congress of Mathematicians, Vol. II, Beijing, 2002, pp. 373–384. Higher Ed. Press, Beijing (2002)

    Google Scholar 

  4. Birman, J.S., Hilden, H.M.: On the mapping class groups of closed surfaces as covering spaces. In: Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969). Ann. of Math. Studies, vol. 66, pp. 81–115 (1971)

    Chapter  Google Scholar 

  5. Birman, J.S., Hilden, H.M.: On isotopies of homeomorphisms of Riemann surfaces. Ann. Math. (2) 97, 424–439 (1973)

    Article  MathSciNet  Google Scholar 

  6. Birman, J.S.: Braids, Links, and Mapping Class Groups. Annals of Mathematics Studies, vol. 82. Princeton University Press, Princeton (1974)

    Google Scholar 

  7. Brendle, T.E., Margalit, D.: Commensurations of the Johnson kernel. Geom. Topol. 8, 1361–1384 (2004)

    Article  MathSciNet  Google Scholar 

  8. Brendle, T.E., Dan, M.: Normal subgroups of mapping class groups and the metaconjecture of Ivanov. J. Am. Math. Soc. 32(4), 1009–1070 (2019)

    Article  MathSciNet  Google Scholar 

  9. Bridson, M.R., Pettet, A., Souto, J.: The abstract commensurator of the Johnson kernels. In preparation

  10. Bridson, M.R., Vogtmann, K.: Automorphism groups of free groups, surface groups and free Abelian groups. In: Problems on Mapping Class Groups and Related Topics. Proc. Sympos. Pure Math., vol. 74, pp. 301–316. Am. Math. Soc., Providence (2006)

    Chapter  Google Scholar 

  11. Bridson, M.R., Wade, R.D.: Direct products of free groups in Aut(FN) (2022). arXiv:2209.01123

  12. Charney, R., Crisp, J.: Automorphism groups of some affine and finite type Artin groups. Math. Res. Lett. 12(2–3), 321–333 (2005)

    Article  MathSciNet  Google Scholar 

  13. Coulon, R.: Outer automorphisms of free Burnside groups. Comment. Math. Helv. 88(4), 789–811 (2013)

    Article  MathSciNet  Google Scholar 

  14. Dyer, J.L., Formanek, E., Grossman, E.K.: On the linearity of automorphism groups of free groups. Arch. Math. (Basel) 38(5), 404–409 (1982)

    Article  MathSciNet  Google Scholar 

  15. Epstein, D.B.A.: Curves on 2-manifolds and isotopies. Acta Math. 115, 83–107 (1966)

    Article  MathSciNet  Google Scholar 

  16. Farb, B., Handel, M.: Commensurations of Out(Fn). Publ. Math. IHES 105(1), 1–48 (2007)

    Article  Google Scholar 

  17. Farb, B., Dan, M.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton (2012)

    Google Scholar 

  18. Hatcher, A.: Homological stability for automorphism groups of free groups. Comment. Math. Helv. 70(1), 39–62 (1995)

    Article  MathSciNet  Google Scholar 

  19. Horbez, C., Wade, R.D.: Commensurations of subgroups of Out(FN). Trans. Am. Math. Soc. 373(4), 2699–2742 (2020)

    Article  Google Scholar 

  20. Ivanov, N.V.: Automorphisms of complexes of curves and Teichmüller spaces. Int. Math. Res. Not. 14, 651–666 (1997)

    Article  Google Scholar 

  21. Mustafa, K.: Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topol. Appl. 95(2), 85–111 (1999)

    Article  MathSciNet  Google Scholar 

  22. Margulis, G.A.: Discrete Subgroups of Semisimple Lie groups. Ergebnisse der Mathematik und Ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17. Springer, Berlin (1991)

    Book  Google Scholar 

  23. Mostow, G.D.: Strong Rigidity of Locally Symmetric Spaces. Annals of Mathematics Studies, vol. 78. Princeton University Press, Princeton (1973)

    Google Scholar 

  24. Dan, M., Winarski, R.R.: Braids groups and mapping class groups: the Birman-Hilden theory. Bull. Lond. Math. Soc. 53(3), 643–659 (2021)

    Article  MathSciNet  Google Scholar 

  25. Stallings, J.R.: Whitehead graphs on handlebodies. In: Geometric Group Theory down Under, Canberra, 1996, pp. 317–330. de Gruyter, Berlin (1999)

    Google Scholar 

  26. Studenmund, D.: Abstract commensurators: a survey of constructions and computations. The Beyond Hyperbolicity Conference Proceedings. To appear

  27. Tits, J.: Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics, vol. 386. Springer, Berlin (1974)

    Google Scholar 

  28. Wade, R.D.: Folding free-group automorphisms. Q. J. Math. 65(1), 291–304 (2014)

    Article  MathSciNet  Google Scholar 

  29. Whitehead, J.H.C.: On certain sets of elements in a free group. Proc. Lond. Math. Soc. (2) 41(1), 48–56 (1936a)

    Article  MathSciNet  Google Scholar 

  30. Whitehead, J.H.C.: On equivalent sets of elements in a free group. Ann. Math. (2) 37(4), 782–800 (1936b)

    Article  MathSciNet  Google Scholar 

  31. Zieschang, H., Vogt, E., Coldewey, H.-D.: Surfaces and Planar Discontinuous Groups. Lecture Notes in Mathematics, vol. 835. Springer, Berlin (1980). Translated from the German by John Stillwell

    Google Scholar 

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Acknowledgements

We thank Mladen Bestvina and Camille Horbez for many fruitful conversations surrounding the material in this paper. We thank Matt Clay for asking about the abstract commensurator of Aut(F2) and for his suggestions concerning its geometry. We also thank the referee for their careful reading and helpful comments. The second author is supported by a University Research Fellowship from the Royal Society.

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  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Martin R. Bridson & Richard D. Wade

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  1. Martin R. Bridson
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Correspondence to Martin R. Bridson.

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Bridson, M.R., Wade, R.D. Commensurations of Aut(FN) and Its Torelli Subgroup. Geom. Funct. Anal. (2024). https://doi.org/10.1007/s00039-024-00681-y

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