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 ℤp[lobrk ]Q[robrk ] for the completed group algebra of Q over ℤp. 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 ℤp[lobrk ]Q[robrk ] to U.