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A Geometric Invariant for Metabelian Pro‐ p Groups
Author(s) -
King Jeremy D.
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799007693
Subject(s) - homomorphism , abelian group , mathematics , finitely generated abelian group , invariant (physics) , metabelian group , combinatorics , g module , algebra over a field , discrete mathematics , pure mathematics , elementary abelian group , rank of an abelian group , mathematical physics
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 .