A Geometric Invariant for Metabelian Pro‐ p Groups

In [ 2 ] Bieri and Strebel introduced a geometric invariant for finitely generated abstract metabelian groups that determines which groups are finitely presented. For a valuable survey of their results, see [ 6 ]; we recall the definition briefly in Section 4. We shall introduce a similar invariant for pro‐ p groups. Let F be the algebraic closure of F p and U be the formal power series algebra F[ T ], with group of units U × . Let Q be a finitely generated abelian pro‐ p group. We write Z p [ Q ] for the completed group algebra of Q over Z p . Let T ( Q ) be the abelian group Hom( Q , U × ) of continuous homomorphisms from Q to U × . We write 1 for the trivial homomorphism. Each v ∈ T ( Q ) extends to a unique continuous algebra homomorphism v̄ from Z p [ Q ] to U .