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

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

2004-01-21

arxiv.org/abs/math/0401269v3

J. Algebra, vol.305 (2006), pp.1093-1101

Mathematics

Group Theory

12 pages, revised version

Scientific paper

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=\phi(u) for some \phi \in AutF_n}. In this paper, we prove that N(u) is bounded by a polynomial function of degree 2n-3 with respect to |u| under the hypothesis that if two letters x, y occur in u, then the total number of x and x^{-1} occurring in u is not equal to the total number of y and y^{-1} occurring in u. We also prove that 2n-3 is the sharp bound on the degree of polynomials bounding N(u). As a special case, we deal with N(u) in F_2 under the same hypothesis.

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