A Geometric Invariant for Metabelian Pro-p Groups

In 2] Bieri and Strebel introduced a geometric invariant forfinitely generated abstract metabelian groups that determineswhich groups are finitely presented. For a valuable survey oftheir results, see 6]; we recall the definition briefly inSection 4. We shall introduce a similar invariant for pro-pgroups. Let F be the algebraic closure of F_p and U be the formal powerseries algebra FT], with group of units U^x. Let Q be a finitelygenerated abelian pro-p group. We write Z_pQ] for the completedgroup algebra of Q over Z_p. Let T(Q) be the abelian group Hom(Q,U^x) of continuous homomorphisms from Q to U^x. We write 1 forthe trivial homomorphism. Each vT(Q) extends to a unique continuousalgebra homomorphism from Z_pQ]to U.