A GEOMETRIC INVARIANT FOR METABELIAN PRO-p GROUPS

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.