FIXED-POINTS OF ENDOMORPHISMS OF A FREE METABELIAN GROUP

We consider IA-endomorphisms (i.e. Identical in Abelianization) of a f ree metabelian group of finite rank, and give a matrix characterizatio n of their fixed points which is similar to (yet different from) the w ell-known characterization of eigenvectors of a linear operator in a v ector space. We then use our matrix characterization to elaborate seve ral properties of the fixed point groups of metabelian endomorphisms. In particular, we show that the rank of the fixed point group of an IA -endomorphism of the free metabelian group of rank n greater than or e qual to 2 can be either equal to 0, 1, or than (n - 1) (in particular, it can be infinite). We also point out a connection between these pro perties of metabelian IA-endomorphisms and some properties of the Gass ner representation of pure braid groups.