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I am looking for a reference for the following result:

Let $\mathbb X$ be a graph of groups whose underlying graph $X$ is finite and whose edge groups are finitely generated. If the fundamental group $\pi_1(\mathbb X,x_0)$ of the graph of groups $\mathbb X$ is itself finitely generated, then so are its vertex groups.

The way I know how to prove this is using folding sequences for graphs of groups (à la Bestvina-Feighn-Stallings-Dunwoody-Kapovich-Weidmann-Myasnikov). Once the machinery is understood it's easy to see. I looked at all these papers but couldn't find this fact stated anywhere. Does anybody know of a reference for this fact?

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Theorem 1.3 of this paper of Haglund and Wise should do it.

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    $\begingroup$ Thanks. I also found this after talking to claude.ai. Even in that paper the result is basically identified as folding sequence folklore. The authors give their own proof that doesn't use folding sequences. $\endgroup$ Commented Mar 25 at 19:11

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