benyash@im.bas-net.by

Submission: 1999, Oct. 4

In the present paper we study the problem of the decomposition of some finitely generated groups into non-trivial free products with amalgamation. Theorem 1 says that a finitely generated group $\Gamma$ is a non-trivial free product with amalgamation if the character variety of irreducible representations of $\Gamma$ into $SL_2(C)$ has dimension more than 1. Theorems 2 and 3 contain results about decomposing of generalized triangle groups into non-trivial free products with amalgamation.One consequence of these theorems is a proof of the conjecture of Fine, Levin, and Rosenberger that any two-generator one-relator group with torsion is a non-trivial free product with amalgamation. As another consequence we obtain that Fuchsian groups $H_1=$ and $H_1=$, $n>1$, are non-trivial free products with amalgamation.

1991 Mathematics Subject Classification: 20E06

Keywords and Phrases: finitely generated group, free products with amalgamation, representation variety, character variety

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