Counting words of minimum length in an automorphic orbit

Details Counting words of minimum length in an automorphic orbit Counting words of minimum length in an automorphic orbit

2003-11-24

arxiv.org/abs/math/0311410v3

J. Algebra, vol.301 (2006), pp.35-58

Mathematics

Group Theory

35 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 \text {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 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. A complete proof without the hypothesis would yield the polynomial time complexity of Whitehead's algorithm for F_n.

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