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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
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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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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