On the palindromic and primitive widths of a free group

Details On the palindromic and primitive widths of a free group On the palindromic and primitive widths of a free group

2003-11-16

arxiv.org/abs/math/0311257v1

Mathematics

Group Theory

10 pages

Scientific paper

Let G be a group and S a subset of G that generates G. For each x in G define the length l_S(x) of x relative to S to be the minimal k such that x is a product of k elements of S. The supremum of the values l_S(x), x \in G, is called the width of G with respect to S. Here we focus on a free group F. The width of F relative to the set of all primitive (respectively palindromic) elements is called the primitive (respectively palindromic) width of F. We prove that for a free group F_n of finite rank n, both widths are infinite. A result of independent interest is that every primitive element of F_2 is a product of at most two palindromes.

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