Dual automorphisms of free groups
Abstract
For any choice of a basis A the free group FN of finite rank N ≥ 2 can be canonically identified with the set F( A) of reduced words in A A-1. However, such a word w ∈ F( A) admits a second interpretation, namely as cylinder C1w ⊂ ∂ FN. The subset of ∂ FN defined by C1w depends not only on the element of FN given by the word w, but also on the chosen basis A. In particular one has in general, for ∈ (FN): (C1w) ≠ C1(w) Indeed, the image of a cylinder under an automorphism ∈ (FN) is in general not a cylinder, but a finite union of cylinders: (C1w)=C1U := ui ∈ U C1ui In his thesis the first author has given an efficient algorithm and a formula how to determine such a (uniquely determined) finite reduced set U = U(w) ⊂ FN. We use those to define the dual automorphism A* by setting A*(w) = U(w). Theorem: For any ∈ (FN) there are at most 2N distinct finite subsets Ui ⊂ FN such that for any w = y1 ... yr ∈ FA there is one of them, say Ui(w), with A*(w) = (w) Ui(w)\, , and Ui(w) depends only on the last letter yr ∈ -1. Furthermore, the seize of each Ui is bounded by 2t, where t ≥ 0 is the number of Nielsen automorphisms in any decomposition of as product of basis permutations, basis inversions and elementary Nielsen automorphisms.