An Automorphism of a Free Metabelian Group Without Fixed Points#

Abstract

We construct an example of an IA-automorphism of the free metabelian group of rank n ≥ 3 without nontrivial fixed points. That gives a negative answer to the question raised by Shpilrain in [Shpilrain, V. (1998). Fixed points of endomorphisms of a free metabelian group. Math. Proc. Cambridge Philos. Soc. 123(1): 75–83. MR 98k:20056]. By a result of Bachmuth [Bachmuth, S. (1965). Automorphisms of free metabelian groups. Trans. Am. Math. Soc. 118:93–1104. MR 31 #4831], such an automorphism does not exist if the rank is equal to 2.