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Feb 11, 2016 at 16:59 comment added YCor You can cook up an example with $H=BS(1,2)^2$ and a well-chosen choice of $B$. Also there are examples where $H$ is Thompson's group $F$.
Feb 11, 2016 at 16:34 comment added user182085 Do you have any example for such HNN- extension? In the example that I found the group $H$ is finitely presented
Feb 11, 2016 at 14:48 comment added YCor I doubt you can get a general answer other than just a restatement. The answer is quite subtle even in the case when $H$ is metabelian (of course, a sufficient condition is that $B$ is finitely presented, but it's not necessary).
Feb 11, 2016 at 14:33 history edited user182085 CC BY-SA 3.0
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Feb 11, 2016 at 14:16 history asked user182085 CC BY-SA 3.0