Counting words of minimum length in an automorphic orbit

Abstract

Let u be a cyclic word in a free group F n of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: |v| = |u| and v = φ(u) for some φ ∈ Aut F n }. In this paper, we prove that N(u) is bounded by a polynomial function with respect to |u| under the hypothesis that if two letters x, y with x = y ±1 occur in u, then the total number of occurrences of x ±1 in u is not equal to the total number of occurrences of y ±1 in u. A complete proof without the hypothesis would yield the polynomial time complexity of Whitehead's algorithm for F n .

An algorithm that decides translation equivalence in a free group of rank two

“…Similarly we denote by n(w; a) the total number of occurrences of a and a −1 in w. Then again clearly n(w; a) = n(w; a −1 ). As in [4], for two automorphisms φ and ψ of F n , by writing φ ≡ ψ we mean the equality of φ and ψ over all cyclic words in F n , that is, φ(w) = ψ(w) for every cyclic word w in F n .…”

Generic properties of Whitehead’s algorithm and isomorphism rigidity of random one-relator groups

“…Substantial further progress for k = 2 has been made by Bilal Khan [2004]. Very interesting partial results regarding the complexity of Whitehead's algorithm for k > 2 have recently been obtained by Donghi Lee [2003].…”

A tighter bound for the number of words of minimum length in an automorphic orbit

“…The same technique as used in [4] is applied to the proofs of these theorems. The proofs will appear in Sections 3-5.…”

“…It is then easy to see that (A, a)(w) = (Ā, a −1 )(w) for any cyclic word w in F n . We also recall the definition of the degree of a Whitehead automorphism of the second type (see [4]): Definition 1.5. Let σ = (A, a) be a Whitehead automorphism of F n of the second type.…”

“…Then, as shown in [4], if x depends on y with respect to w, then y depends on x with respect to w. We then construct the dependence graph Γ w of w as follows: Take the vertex set as Σ, and connect two distinct vertices x, y ∈ Σ by a non-oriented edge if either y = x −1 or y depends on x with respect to w. Let C i be the connected component of Γ w containing x i . Clearly there exists a unique factorization…”