Decomposing some finitely generated groups into free products with amalgamation

In this paper we study the problem of decomposing finitely generated groups into non-trivial free products with amalgamation. We prove that if $\dim X^s(G)>1$, where $X^s(G)$ is the character variety of irreducible representations of G into $SL_2(C)$, then G is a non-trivial free product with amalgamation. We also consider a generalized triangle group $G=<a,b | a^n= b^k=R^m(a,b)>$. It is proved that if one of the generators of G has an infinite order, then G is a non-trivial free product with amalgamation. In general case we find some sufficient conditions under which G is a non-trivial free product with amalgamation