Decomposing finitely generated groups into free products with amalgamation (Q2759299)

The problem of the existence of a decomposition of a finitely generated group \(\Gamma\) into a non-trivial free product with amalgamation is studied. The following main results are established: (1) If the character variety \(X^S(\Gamma)\) of irreducible representations of \(\Gamma\) into \(\text{SL}_2(\mathbb{C})\) satisfies \(\dim X^S(\Gamma)\geq 2\), then \(\Gamma\) is a non-trivial free product with amalgamation (Theorem 1). (2) In Theorem 2, the author considers the case of a generalized triangle group \(\Gamma_n=\langle a,b:a^n=b^k=R^m(a,b)=1\rangle\). (3) \(\Gamma_n\) is considered with \(k=0\) and \(n=0\) or \(n\geq 2\), \(m\geq 2\) (Theorem 3). This Theorem gives as a consequence the proof of a conjecture due to Fine, Levin and Rosenberger that if \(\langle a,b:R^m(a,b)=1\rangle\), \(m\geq 2\) is a group with torsion then it is a non-trivial free product with amalgamation.