1996/05/17 by Vladimir Shpilrain, Shpilrain, Vladimir
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR
paper · pdf · doi:10.48550/arxiv.math/9605206
LaTex file, 11 pages
arxiv created 1996/05/17 · openalex publication_date 1996/05/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider IA-endomorphisms (i.e., Identical in Abelianization) of a free metabelian group of finite rank, and give a matrix characterization of their fixed 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 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 ≥ 2 can be either equal to 0, 1, or greater than (n-1) (in particular, it can be infinite). We also point out a connection between these properties of metabelian IA-endomorphisms and some properties of the Gassner representation of pure braid groups.