Decomposing finitely generated groups into free products with amalgamation

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