Fixed points of endomorphisms of a free metabelian group

Abstract

We consider IA-endomorphisms (i.e. Identical in Abelianization) of a free metabelian group of nite rank, and give a matrix characterization of their xed points which is similar to (yet different from) the well-known characterization of eigenvectors of a linear operator in a vector space. We then use our matrix characterization to elaborate several properties of the xed point groups of metabelian endomorphisms. In particular, we show that the rank of the xed point group of an IA-endomorphism of the free metabelian group of rank n> 2 can be either equal to 0, 1, or greater than (n 1) (in particular, it can be innite). We also point out a connection between these properties of metabelian IA-endomorphisms and some properties of the Gassner representation of pure braid groups.