My curiosity was raised by the [following question](https://mathoverflow.net/questions/104400/when-is-an-hnn-extension-finitely-presented)
and the huge variety of comments and suggestions it attracted. I wondered if a converse statement might be equally interesting.


Let $G$ be a finitely presented group which admits a non-trivial graph of groups decomposition. Does it follow that each vertex stabiliser is finitely presented? How about if every edge stabiliser is finite?


[Dunwoody](http://link.springer.com/article/10.1007%2FBF01388581#page-1) comments that if $G$ is finitely generated and each edge stabiliser is finite then each vertex stabiliser is finitely generated.



Finiteness properties for graph of groups decompositions
Does it follow that each vertex stabiliser is finitely presented: No.


Just notice that there exist infinitely presented groups $H$ with an automorphism such that the corresponding semidirect product $G=H\rtimes\mathbf{Z}$ is finitely presented. This is a non-trivial graph of groups decomposition, namely an HNN-decomposition where the vertex group and edge group coincide; in case you find this too degenerate, just add some further flourish to the graph of groups, so that $H$ still appears as a vertex group and the resulting group is still finitely presented. One example is when $G=BS(1,2)^2=\mathbf{Z}[1/2]^2\rtimes\mathbf{Z}^2$ where the generators of $\mathbf{Z}$ act by the diagonal matrices $(2,1)$ and $(1,2)$, and $H=\mathbf{Z}[1/2]^2\rtimes\mathbf{Z}$ where $\mathbf{Z}$ acts by the diagonal matrix $(2,1/2)$.


On the other hand, for the second question "How about if every edge stabiliser is finite?", the answer is yes. Indeed, write $H$ as a limit of truncated presentations $H\_n$ (this limit here means: inductive limit of surjections between f.g. groups). For $n$ large enough (say $n\ge n\_0$), all finite subgroups of $G$ occurring in the graph of groups decomposition of $G$ are well-defined, and this allows to write $G$ as (non-attained) limit of corresponding graph of groups $G\_n$ for $n\ge 0$, where in the graph of groups, $H$ was replaced with $H\_n$). The argument still works if the edge groups are finitely presented instead of finite.


