Abstract
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 [ ] be the algebraic closure of [ ] p and U be the formal power series algebra [ ][lobrk ] T [robrk ], with group of units U × . Let Q be a finitely generated abelian pro- p group. We write ℤ p [lobrk ] Q [robrk ] 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 vmacr; from ℤ p [lobrk ] Q [robrk ] to U .
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