Amalgamated products and finitely presented groups (Q923128)

An amalgamated product \(A*_ CB\) is said to be non-trivial if the amalgamated subgroup C is not equal to either of the factors A, B. If, in addition, \(\max \{| A:C|,| B:C| \}>2\), then the amalgamated product is said to be proper. The authors first establish the following theorem (they point out that the theorem has been obtained independently by \textit{M. Bestvina} and \textit{M. Feighn} [unpublished]); If a finitely presented group G has a proper amalgamated product decomposition \(A*_ CB\), then it has a proper decomposition \(A'*_{C'}B'\) where \(A'\leq A\), \(B'\leq B\), \(C'\leq C\) and where \(A'\), \(B'\), \(C'\) are finitely generated (Theorem 1). They then make clever use of this result to obtain two further interesting theorems (Theorems 2, 3). Let \(G=<x_ 1,...,x_ m;R_ 1,...,R_ n>\) with \(m\geq 2\), and suppose that either m-n\(\geq 2\) (Theorem 2), or \(R_ 1,...,R_ n\) lie in the third derived subgroup of the free group on \(x_ 1,...,x_ m\) (Theorem 3). Then G has a proper amalgamated product decomposition in which the factors and amalgamated subgroup are finitely generated.