Decomposing finitely generated groups into free products with amalgamation

The problem of the existence of a decomposition of a finitely generated group $\Gamma$ into a non-trivial free product with amalgamation is studied. It is proved that if $\dim X^s(\Gamma )\geqslant 2$, where $X^s(\Gamma )$ is the character variety of irreducible representations of $\Gamma$ into $\operatorname {SL}_2(\mathbb C)$, then $\Gamma$ is a non-trivial free product with amalgamation. Next, the case when $\Gamma =\langle a,b\mid a^n=b^k=R^m(a,b)\rangle $ is a generalized triangle group is considered. It is proved that if one of the generators of $\Gamma$ has infinite order, then $\Gamma$ is a non-trivial free product with amalgamation. In the general case sufficient conditions ensuring that $\Gamma$ is a non-trivial free product with amalgamation are found.