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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Classes of automorphisms of free groups of infinite rank


Author: Robert Cohen
Journal: Trans. Amer. Math. Soc. 177 (1973), 99-120
MSC: Primary 20F55
DOI: https://doi.org/10.1090/S0002-9947-1973-0316581-0
MathSciNet review: 0316581
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Abstract: This paper is concerned with finding classes of automorphisms of an infinitely generated free group F which can be generated by ``elementary'' Nielsen transformations. Two different notions of ``elementary'' Nielsen transformations are explored. One leads to a classification of the automorphisms generated by these transformations. The other notion leads to the subgroup B of $ {\operatorname{Aut}}(F)$ consisting of the ``bounded length'' automorphisms of F. We prove that the class of ``bounded 3-length'' automorphisms $ {B_3}$ and the class of ``elementary simultaneous'' Nielsen transformations generate the same subgroup of $ {\operatorname{Aut}}(F)$. We show that for the class T of automorphisms of ``2 occurring generators", the groups generated by $ T \cap B$ and the ``elementary simultaneous'' Nielsen transformations are identical. These results lead to the conjecture that B is generated by the ``elementary simultaneous Nielsen transformations".

A study is also made of the subgroup S of the ``triangular automorphisms'' of $ {F_\infty }$, the free group on a countably infinite set of free generators. It is found that a ``triangular automorphism'' may be factored into ``splitting automorphisms'' of $ {F_\infty }$, which may be viewed as the ``elementary'' automorphisms of S.


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  • [1] M. Hall, Jr., The theory of groups, Macmillan, New York, 1959. MR 21 #1996. MR 0103215 (21:1996)
  • [2] F. W. Levi, Über die Untergruppen der freien Gruppen, Math. Z. 32 (1930), 315-318. MR 1545167
  • [3] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Pure and Appl. Math., vol. 13, Interscience, New York, 1966. MR 34 #7617. MR 0207802 (34:7617)
  • [4] J. Nielsen, Om Regnig med ikke-kommutative Faktorer og dens Anvendeise i Gruppeteorien, Mat. Tidsskr. B 1921, 77-94.
  • [5] -, Die Isomorphismengruppe der freien Gruppen, Math. Ann. 91 (1924), 169-209. MR 1512188
  • [6] E. S. Rapaport, On free groups and their automorphisms, Acta Math. 99 (1958), 139-163. MR 24 #A1302. MR 0131452 (24:A1302)
  • [7] D. H. Wagner, On free products of groups, Trans. Amer. Math. Soc. 84 (1957), 352-378. MR 18, 790. MR 0083992 (18:790a)
  • [8] J. H. C. Whitehead, On certain sets of elements in a free group, Proc. London Math. Soc. 41 (1936), 48-56.
  • [9] -, On equivalent sets of elements in a free group, Ann. of Math. 37 (1936), 782-800. MR 1503309

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0316581-0
Keywords: Free group, automorphism, Nielsen reduction, elementary Nielsen transformation, elementary simultaneous Nielsen transformation, triangular automorphism, splitting automorphism, bounded automorphism
Article copyright: © Copyright 1973 American Mathematical Society

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