Skip to main content
SpringerLink
Log in

Actions of \(\mathbf {SAut}\varvec{(F_n)}\)

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We study actions of SAut\((F_n)\), the unique subgroup of index two in the automorphism group of a free group of rank n, and obtain rigidity results for its representations. In particular, we show that every smooth action of SAut\((F_n)\) on a low dimensional torus is trivial.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Bridson and K. Vogtmann, Actions of automorphism groups of free groups on homology spheres and acyclic manifolds, Comment. Math. Helv. 86 (2011), 73–90.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Bridson and K. Vogtmann, Homomorphisms from automorphism groups of free groups, Bull. London Math. Soc. 35 (2003), 785–792.

    Article  MathSciNet  MATH  Google Scholar 

  3. W. Burnside, Theory of Groups of Finite Order, Dover Publications, Inc., New York, 1955.

    MATH  Google Scholar 

  4. K. Hoffmann and R.  Kunze, Linear Algebra, 2nd edition. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1971.

    Google Scholar 

  5. R. A. Horn and C. R. Johnson, Matrix Analysis, Paperback edition, Cambridge University Press, New York, 1990.

  6. G. James and M. Liebeck, Representations and Characters of Groups, 2nd edition, Cambridge University Press, New York, 2001.

    Book  MATH  Google Scholar 

  7. D. Kielak, Outer automorphism groups of free groups: linear and free representations, J. Lond. Math. Soc. 87 (2013), 917–942.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Potapchik and A. Rapinchuk, Low-dimensional linear representations of Aut\((F_n)\), \(n\ge 3\), Trans. Amer. Math. Soc. 352 (2000), 1437-1451.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Wagner, The faithful linear representation of least degree of \(S_n\) and \(A_n\) over a field of characteristic 2, Math. Z. 151 (1976), 127–137.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Wagner, The faithful linear representations of least degree of \(S_n\) and \(A_n\) over a field of odd characteristic, Math. Z. 154 (1977), 103–114.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Weinberger, \({{\rm SL}}_n({\mathbb{Z}})\) cannot act on small tori, In: Geometric Topology (Athens, GA, 1993), 406–408, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Münster University, Einsteinstrasse 62, 48149, Münster, Germany

    Olga Varghese

Authors
  1. Olga Varghese
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olga Varghese.

Additional information

Research partially supported by SFB 878.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Varghese, O. Actions of \(\mathbf {SAut}\varvec{(F_n)}\). Arch. Math. 110, 319–325 (2018). https://doi.org/10.1007/s00013-017-1138-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-017-1138-9

Keywords

Mathematics Subject Classification

Search

Navigation