Classes of automorphisms of free groups of infinite rank

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 Aut ( F ) {\operatorname {Aut}}(F) consisting of the “bounded length” automorphisms of F. We prove that the class of “bounded 3-length” automorphisms B 3 {B_3} and the class of “elementary simultaneous” Nielsen transformations generate the same subgroup of Aut ( F ) {\operatorname {Aut}}(F) . We show that for the class T of automorphisms of “2 occurring generators", the groups generated by T ∩ B 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 ∞ {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 ∞ {F_\infty } , which may be viewed as the “elementary” automorphisms of S.