, Volume 99, Issue 5, pp 425-432,
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Fixed points of automorphisms preserving the length of words in free solvable groups

Abstract

Let δ be an automorphism of prime order p of the free group F n . Suppose δ has no fixed points and preserves the length of words. By σ := δ (m) we denote the automorphism of the free solvable group ${F_{n}/F_n^{(m)} }$ induced by δ. We show that every fixed point of σ has the form ${cc^{\sigma} \ldots c^{\sigma^{p-1}}}$ , where ${c\in F_n^{(m-1)}/F_n^{(m)}}$ . This is a generalization of some known results, including the Macedońska–Solitar Theorem [10].