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It is clear that cyclic groups have the property that they cannot be written in a non-trivial way as an amalgamated free product or as an HNN-extension.

Can someone please provide us examples of torsion-free 2-generated groups having this property? Any comments related to this question are welcome!

[I'm aware of Serre's Theorem 15 in Trees.]


Edit: The term "in a non-trivial way" means the following. For an amalgamated free product $G=H*_C K$ it is required $H\ne C\ne K$. For an HNN-extension $G=HNN(H,A,B,t)$ it is required $A\ne H\ne B$.

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    Nilo, if you insist that $A\neq H\neq B$ then you're not just allowing $\mathbb{Z}$, but also ascending HNN-extensions, including Thompson's group F (if I recall correctly). In your comment below my answer, you seemed to want to exclude Thompson's group F. You can't have it both ways.2011-01-25
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    You are definitely right.2011-01-25
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    So what's the answer? Are you looking for a two-generator torsion-free group that doesn't split non-trivially in the usual sense, or for one that doesn't split non-trivially in your sense? Because if the latter, then $\mathbb{Z}^2$ works.2011-01-25
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    Thanks, Henry! I missed that completely. I would be glad to know what is the usual sense... Thanks again.2011-01-26
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    I've added an explicit example to my answer below.2011-01-27

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