Algebraic & Geometric Topology

Intersections of automorphism fixed subgroups in the free group of rank three

Armando Martino

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Abstract

We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism.

Article information

Source
Algebr. Geom. Topol., Volume 4, Number 1 (2004), 177-198.

Dates
Received: 30 December 2002
Revised: 22 March 2004
Accepted: 25 March 2004
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882473

Digital Object Identifier
doi:10.2140/agt.2004.4.177

Mathematical Reviews number (MathSciNet)
MR2059188

Zentralblatt MATH identifier
1054.20009

Subjects
Primary: 20E05: Free nonabelian groups
Secondary: 20F28: Automorphism groups of groups [See also 20E36]

Keywords
free group automorphism fixed subgroup

Citation

Martino, Armando. Intersections of automorphism fixed subgroups in the free group of rank three. Algebr. Geom. Topol. 4 (2004), no. 1, 177--198. doi:10.2140/agt.2004.4.177. https://projecteuclid.org/euclid.agt/1513882473


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References

  • G.M. Bergman, Supports of derivarions, free factorizations and ranks of fixed subgroups in free groups, Trans. Amer. Math. Soc., 351 (1999), 1531-1550.
  • M. Bestvina, M. Feighn and M. Handel, The Tits Alternative for $Out(F_n)$ I: Dynamics of Exponentially Growing Automorphisms, Annals of Math. 151 (2000), 517-623.
  • Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for ${\rm Out}({F}_n)$. II. A Kolchin theorem.
  • M. Bestvina, M. Handel, Train tracks and automorphisms of free groups, Ann. of Math., 135 (1992), 1-51.
  • D.J. Collins, E.C. Turner, All automorphisms of free groups with maximal rank fixed subgroups, Math. Proc. Cambridge Philos. Soc., 119 (1996), 615-630.
  • W. Dicks, E. Ventura, The group fixed by a family of injective endomorphism of a free group, Contemp. Math., 195 (1996), 1-81.
  • J.L. Dyer, G.P. Scott, Periodic automorphisms of free groups, Comm. Alg., 3 (1975), 195-201.
  • W. Imrich, E.C. Turner, Endomorphisms of free groups and their fixed points, Math. Proc. Cambridge Philos. Soc., 105 (1989), 421-422.
  • I. Kapovich and A. Myasnikov, Stallings Foldings and Subgroups of Free Groups, J. Algebra, 248, 2 (2002), 608-668.
  • R.C. Lyndon and P.E. Schupp, “Combinatorial group theory", Springer-Verlag, Berlin, 1977.
  • W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, Interscience Publishers, New York, (1966).
  • A. Martino, E. Ventura, A description of auto-fixed subgroups in the free group, to appear in Topology.\qua http://www-eupm.upc.es/~ventura/
  • A. Martino, E. Ventura, Fixed subgroups are compressed in free groups, to appear Comm Alg.
  • A. Martino, E. Ventura, On automorphism-fixed subgroups of a free group, J. Algebra, 230 (2000), 596-607.
  • John R. Stallings, Whitehead graphs on handlebodies, Geometric group theory down under (Canberra, 1996), de Gruyter, Berlin (1999), 317–330.
  • E.C. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc., 28 (1996), 255-263.
  • E. Ventura, On fixed subgroups of maximal rank, Comm. Algebra, 25 (1997), 3361-3375.
  • J.H.C. Whitehead, On certain sets of elements in a free group, Proc. London Math. Soc., 41 (1936), 48-56.